Classically, Grothendieck's motives are only the pure motives, meaning abelian-ish things which capture the (Weil-cohomology-style) $H^i$ of smooth, projective varieties. To see the relationship with motivic cohomology, one should extend the notion of motive so that non-pure (i.e. "mixed") motives are allowed, these mixed motives being abelian-ish things which capture the $H^i$ of arbitrary varieties. The main novelty with mixed motives is that the (conjectural) abelian category of them is not semi-simple -- in fact every mixed motive should be a (generally non-trivial) iterated extension of pure motives, these extensions essentially coming from compactification and resolution of singularities, as in the story of mixed Hodge structures.
Then once one thinks of mixed motives, a natural direction of study (or speculation, as the case may be...) is that of determining all possible extensions (or iterated extensions) between two motives. And that's what motivic cohomology is, essentially: the study of these Ext groups. More formally, every variety $X$ should determine an object $C(X)$ in the bounded derived category of mixed motives, collecting together all the various mixed motives $H^i(X)$, and the $(i,j)^{th}$ motivic cohomology of $X$ is (up to twisting conventions) the abelian group of maps from the unit object to $C(X)$ \ $[i](j)$ (the $j^{th}$ Tate twist of the $i^{th}$ shift of $C(X)$) in the derived category of mixed motives.
Now, there are a few points to make here. The first is that, though the above motivation and definition of motivic cohomology rely on an as-yet-conjectural abelian category of mixed motives, one can, independently of any conjectures, define a triangulated category which, as far as anyone can tell, behaves as if it were the bounded derived category of this conjectural abelian category. The most popular such definition, because of its simplicity and relative workability, is Voevodsky's. So the basic theory and many basic results on motivic cohomology are unconditional.
Another thing to say is that, as always, matters with motives are illuminated by considering realization functors. Let me single out the $\ell$-adic etale realization, since its extension from pure to mixed motives is straightforward (unlike for Hodge structures): any mixed motive, just as any pure motive, yields a finite-dimensional $\ell$-adic vector space with a continuous action of the absolute Galois group of our base field. It then "follows" (in our conjectural framework... or actually follows, without quotation marks, in Voevodsky's framework) that the $(i,j)^{th}$ motivic cohomology of X maps to the abelian group of maps from the unit object to $C^{et}(X)$ \ $[i](j)$ in the bounded derived category of $\ell$-adic Galois representations. But this abelian group of maps is just the classical (continuous) $\ell$-adic etale cohomology $H^i(X(j))$ of the variety $X$, making this latter group the natural target of an $\ell$-adic etale "realization" map from motivic cohomology.
So here comes the third point: note that this is the etale cohomology of $X$ itself, not of the base change from $X$ to its algebraic closure. So this etale cohomology group mixes up arithmetic information and geometric information, and the same is true of motivic cohomology in general. (Think especially of the case $X=pt$: the motivic cohomology of a point admits a generally nontrivial realization map to the $\ell$-adic Galois cohomology of the base field.) For example, it is expected (e.g. by Grothendieck -- see http://www.math.jussieu.fr/~leila/grothendieckcircle/motives.pdf for this and more) that for an abelian variety $A$ over an ``arithmetic'' base field $k$, the most interesting part of the motivic cohomology $H^{(2,1)}(A)$ (again my twists may be off...), by which I mean the direct summand which classifies extensions of $H^1(A)$ by $H^1(G_m)$, should identify with the rationalization of the abelian group of $k$-rational points of the dual abelian variety of $A$, the map being given by associating to such $k$-rational point the mixed motive given as $H^1$ of the total space of the corresponding $G_m$-torsor on $A$. And in this case, the above "realization" map to $\ell$-adic etale cohomology is closely related to the classical Kummer-style map used in the proof of the Mordell-Weil theorem.
So in a nutshell: motivic cohomology is very related to motives, since morally it classifies extensions of motives. But it is of a different nature, since it is an abelian group rather than an object of a more exotic abelian category; and it's also quite different from a human standpoint in that we know how to define it unconditionally. Finally, motivic cohomology realizes to Galois cohomology of a variety itself, rather than to the base change of such a variety to the algebraic closure.
Hope this was helpful in some way.
Best Answer
Let me talk about Beilinson's conjectures by beginning with $\zeta$-functions of number fields and $K$-theory. Space is limited, but let me see if I can tell a coherent story.
The Dedekind zeta function and the Dirichlet regulator
Suppose $F$ a number field, with $$[F:\mathbf{Q}]=n=r_1+2r_2,$$ where $r_1$ is the number of real embeddings, and $r_2$ is the number of complex embeddings. Write $\mathcal{O}$ for the ring of integers of $F$.
Here's the power series for the Dedekind zeta function: $$\zeta_F(s)=\sum|(\mathcal{O}/I)|^{-s},$$ where the sum is taken over nonzero ideals $I$ of $\mathcal{O}$.
Here are a few key analytical facts about this power series:
This power series converges absolutely for $\Re(s)>1$.
The function $\zeta_F(s)$ can be analytically continued to a meromorphic function on $\mathbf{C}$ with a simple pole at $s=1$.
There is the Euler product expansion: $$\zeta_F(s)=\prod_{0\neq p\in\mathrm{Spec}(\mathcal{O}_F)}\frac{1}{1-|(\mathcal{O}_F/p)|^{-s}}.$$
The Dedekind zeta function satisfies a functional equation relating $\zeta_F(1-s)$ and $\zeta_F(s).$
If $m$ is a positive integer, $\zeta_F(s)$ has a (possible) zero at $s=1-m$ of order $$d_m=\begin{cases}r_1+r_2-1&\textrm{if }m=1;\\ r_1+r_2&\textrm{if }m>1\textrm{ is odd};\\ r_2&\textrm{if }m>1\textrm{ is even}, \end{cases}$$ and its special value at $s=1-m$ is $$\zeta_F^{\star}(1-m)=\lim_{s\to 1-m}(s+m-1)^{-d_m}\zeta_F(s),$$ the first nonzero coefficient of the Taylor expansion around $1-m$.
Our interest is in these special values of $\zeta_F(s)$ at $s=1-m$. At the end of the 19th century, Dirichlet discovered an arithmetic interpretation of the special value $\zeta_F^{\star}(0)$. Recall that the Dirichlet regulator map is the logarithmic embedding $$\rho_F^D:\mathcal{O}_F^{\times}/\mu_F\to\mathbf{R}^{r_1+r_2-1},$$ where $\mu_F$ is the group of roots of unity of $F$. The covolume of the image lattice is the the Dirichlet regulator $R^D_F$. With this, we have the
Dirichlet Analytic Class Number Formula. The order of vanishing of $\zeta_F(s)$ at $s=0$ is $\operatorname{rank}_\mathbf{Z}\mathcal{O}_F^\times$, and the special value of $\zeta_F(s)$ at $s=0$ is given by the formula $$\zeta_F^{\star}(0)=-\frac{|\mathrm{Pic}(\mathcal{O}_F)|}{|\mu_F|}R^D_F.$$
Now, using what we know about the lower $K$-theory, we have: $$K_0(\mathcal{O})\cong\mathbf{Z}\oplus\mathrm{Pic}(\mathcal{O})$$ and $$K_1(\mathcal{O}_F)\cong\mathcal{O}_F^{\times}.$$
So the Dirichlet Analytic Class Number Formula reads: $$\zeta_F^{\star}(0)=-\frac{|{}^{\tau}K_0(\mathcal{O})|}{|{}^{\tau}K_1(\mathcal{O})|}R^D_F,$$ where ${}^{\tau}A$ denotes the torsion subgroup of the abelian group $A$.
The Borel regulator and the Lichtenbaum conjectures
Let us keep the notations from the previous section.
Theorem [Borel]. If $m>0$ is even, then $K_m(\mathcal{O})$ is finite.
In the early 1970s, A. Borel constructed the Borel regulator maps, using the structure of the homology of $SL_n(\mathcal{O})$. These are homomorphisms $$\rho_{F,m}^B:K_{2m-1}(\mathcal{O})\to\mathbf{R}^{d_m},$$ one for every integer $m>0$, generalizing the Dirichlet regulator (which is the Borel regulator when $m=1$). Borel showed that for any integer $m>0$ the kernel of $\rho_{F,m}^B$ is finite, and that the induced map $$\rho_{F,m}^B\otimes\mathbf{R}:K_{2m-1}(\mathcal{O})\otimes\mathbf{R}\to\mathbf{R}^{d_m}$$ is an isomorphism. That is, the rank of $K_{2m-1}(\mathcal{O})$ is equal to the order of vanishing $d_m$ of the Dedekind zeta function $\zeta_F(s)$ at $s=1-m$. Hence the image of $\rho_{F,m}^B$ is a lattice in $\mathbf{R}^{d_m}$; its covolume is called the Borel regulator $R_{F,m}^B$.
Borel showed that the special value of $\zeta_F(s)$ at $s=1-m$ is a rational multiple of the Borel regulator $R_{F,m}^B$, viz.: $$\zeta_F^{\star}(1-m)=Q_{F,m}R_{F,m}^B.$$ Lichtenbaum was led to give the following conjecture in around 1971, which gives a conjectural description of $Q_{F,m}$.
Conjecture [Lichtenbaum]. For any integer $m>0$, one has $$|\zeta_F^{\star}(1-m)|"="\frac{|{}^{\tau}K_{2m-2}(\mathcal{O})|}{|{}^{\tau}K_{2m-1}(\mathcal{O})|}R_{F,m}^B.$$ (Here the notation $"="$ indicates that one has equality up to a power of $2$.)
Beilinson's conjectures
Suppose now that $X$ is a smooth proper variety of dimension $n$ over $F$; for simplicity, let's assume that $X$ has good reduction at all primes. The question we might ask is, what could be an analogue for the Lichtenbaum conjectures that might provide us with an interpretation of the special values of $L$-functions of $X$? It turns out that since number fields have motivic cohomological dimension $1$, special values of their $\zeta$-functions can be formulated using only $K$-theory, but life is not so easy if we have higher-dimensional varieties; for this, we must use the weight filtration on $K$-theory in detail; this leads us to motivic cohomology.
Write $\overline{X}:=X\otimes_F\overline{F}$. Now for every nonzero prime $p\in\mathrm{Spec}(\mathcal{O})$, we may choose a prime $q\in\mathrm{Spec}(\overline{\mathcal{O}})$ lying over $p$, and we can contemplate the decomposition subgroup $D_{q}\subset G_F$ and the inertia subgroup $I_{q}\subset D_{q}$.
Now if $\ell$ is a prime over which $p$ does not lie and $0\leq i\leq 2n$, then the inverse $\phi_{q}^{-1}$ of the arithmetic Frobenius $\phi_{q}\in D_{q}/I_{q}$ acts on the $I_{q}$-invariant subspace $H_{\ell}^i(\overline{X})^{I_{q}}$ of the $\ell$-adic cohomology $H_{\ell}^i(\overline{X})$. We can contemplate the characteristic polynomial of this action: $$P_{p}(i,x):=\det(1-x\phi_{q}^{-1}).$$ One sees that $P_{p}(i,x)$ does not depend on the particular choice of $q$, and it is a consequence of Deligne's proof of the Weil conjectures that the polynomial $P_{p}(i,x)$ has integer coefficients that are independent of $\ell$. (If there are primes of bad reduction, this is expected by a conjecture of Serre.)
This permits us to define the local $L$-factor at the corresponding finite place $\nu(p)$: $$L_{\nu(p)}(X,i,s):=\frac{1}{P_{p}(i,p^{-s})}$$ We can also define local $L$-factors at infinite places as well. For the sake of brevity, let me skip over this for now. (I can fill in the details later if you like.)
With these local $L$-factors, we define the $L$-function of $X$ via the Euler product expansion $$L(X,i,s):=\prod_{0\neq p\in\mathrm{Spec}(\mathcal{O})}L_{\nu(p)}(X,i,s);$$ this product converges absolutely for $\Re(s)\gg 0$. We also define the $L$-function at the infinite prime $$L_{\infty}(X,i,s):=\prod_{\nu|\infty}L_{\nu}(X,i,s)$$ and the full $L$-function $$\Lambda(X,i,s)=L_{\infty}(X,i,s)L(X,i,s).$$
Here are the expected analytical properties of the $L$-function of $X$.
The Euler product converges absolutely for $\Re(s)>\frac{i}{2}+1$.
$L(X,i,s)$ admits a meromorphic continuation to the complex plane, and the only possible pole occurs at $s=\frac{i}{2}+1$ for $i$ even.
$L\left(X,i,\frac{i}{2}+1\right)\neq 0$.
There is a functional equation relating $\Lambda(X,i,s)$ and $\Lambda(X,i,i+1-s).$
Beilinson constructs the Beilinson regulator $\rho$ from the part $H^{i+1}_{\mu}(\mathcal{X},\mathbf{Q}(r))$ of rational motivic cohomology of $X$ coming from a smooth and proper model $\mathcal{X}$ of $X$ (conjectured to be an invariant of the choice of $\mathcal{X}$) to Deligne-Beilinson cohomology $D^{i+1}(X,\mathbf{R}(r))$. This has already been discussed here. It's nice to know that we now have a precise relationship between the Beilinson regulator and the Borel regulator. (They agree up to exactly the fudge factor power of $2$ that appears in the statement of the Lichtenbaum conjecture above.)
Let's now assume $r<\frac{i}{2}$.
Conjecture [Beilinson]. The Beilinson regulator $\rho$ induces an isomorphism $$H^{i+1}_{\mu}(\mathcal{X},\mathbf{Q}(r))\otimes\mathbf{R}\cong D^{i+1}(X,\mathbf{R}(r)),$$ and if $c_X(r)\in\mathbf{R}^{\times}/\mathbf{Q}^{\times}$ is the isomorphism above calculated in rational bases, then $$L^{\star}(X,i,r)\equiv c_X(r)\mod\mathbf{Q}^{\times}.$$