In general, for congruence subgroups $\Gamma$ of $SL(2,\mathbb Z)$, the collection of weight $k$ Eisenstein series is spanned by functions formed by
$$ E(z)=E(z,c_o,d_o,N,k)=\sum''_{c=c_o,d=d_o(N)}\;\;\frac{1}{(cz+d)^k}$$
where $c,d$ run over _relatively_prime_ integers congruence to $c_o,d_o$ mod $N$. Especially for composite $N$, there are (elementary) relations among these. Eisenstein series are far more elementary objects than most other modular forms.
However, those sums only converge absolutely for $k>2$. Many decades ago, E. Hecke had already addressed this "problem", by what is now called "Hecke summation" (although it can be understood more systematically, too): throw in a "summation factor" to make the series converge when $k=2$:
$$ E(z,s) = \sum''_{c=c_o,d=d_o(N)}\;\frac{1}{|cz+d|^{2s}\cdot (cz+d)^2}$$
with $\Re(s)>0$. The plan is to analytically continue in $s$ to a neighborhood of $s=0$, and then set $s=0$.
The latter plan does largely succeed, except that it does not always produce holomorphic modular forms at $s=0$. This "disappointment" already occurs for $\Gamma=SL(2,\mathbb Z)$, where there is an extra summand of $1/y$ as part of the $0$th Fourier coefficient of the Hecke-summed Eisenstein series.
For Hilbert modular forms, that is, the analogue of elliptic modular forms but over larger (totally real) number fields in place of $\mathbb Q$, the "disappointment" never occurs. Although the computation to see the outcome is essentially elementary, the reason for it is not so elementary.
The general idea is that, given a subgroup $\Gamma$ of finite index inside $SL_2(\mathbb Z)$ (though also we'd often require that this subgroup be definable by "congruence conditions" on the entries), the quotient $X=\Gamma\backslash{\mathfrak H}$ of the upper half-plane $\mathfrak H$ by $\Gamma$ needs finitely-many points added to it to "compactify" it. These are the "cusps". With fixed $\Gamma$ and fixed "weight" $k$, the cuspforms are the holomorphic weight-$k$ modular forms "vanishing at all cusps". (Since holomorphic modular forms are not actually invariant by $\Gamma$, this notion of vanishing includes some technicalities...) For even weight $2k>2$, the dimension of the space of weight-$2k$ holomorphic modular forms modulo cuspforms is equal to the number of cusps. (For odd weight $2k+1$, depending on $\Gamma$, some cusps can be "irregular", or some other modifier, in the sense of admitting no non-vanishing holomorphic modular form...) Thus, at least for even weight, relative to fixed $\Gamma$, there is an Eisenstein series attached to each cusp, which takes non-zero value there, and value $0$ at all other cusps.
How to exhibit/construct these? The action of $\Gamma$ extends to the compactification, and the isotropy (=stabilizer) subgroup $\Gamma_\sigma$ of a given cusp $\sigma$ makes sense. The corresponding Eisenstein series is a sum over $\Gamma_\sigma\backslash \Gamma$... For $\Gamma=SL_2(\mathbb Z)$ there is a single (equivalence class of) cusp, $i\infty$, and the expression written in the question is one formulaic version of the corresponding formation of Eisenstein series as sum over a coset space of this type.
The dimensions of spaces of holomorphic cuspforms are computable via Riemann-Roch, in effect. This is subtler than computing the dimensions of spaces of holomorphic Eisenstein series.
Best Answer
The question has some implicit hypotheses, possibly not clear to the questioner, and this implicitness and ambiguities about it complicate matters. First, the more natural descriptions of Eisenstein series for GL(2) of weights k>2 are not of the form in the question, but are $\sum_{c,d} 1/(cz+d)^k$. This does not converge for $k=2$, so $k=2$ has to be approached more delicately (via an analytic continuation, Hecke summation, producing _something_like_ the expression in the question). However, at level one, that is, for $SL_2(\mathbb Z)$, there is no truly-holomorphic Eisenstein series of weight $2$. The analytic continuation has an extra term, which one may discard, but then destroying the literal automorphy condition. Maybe that doesn't matter, but one should be careful about "understandings".
Thus, depending what one means, wants, or needs, while at higher levels the meromorphic continuation can produce holomorphic modular forms at level 2. (This positive outcome always occurs for Hilbert modular forms, that is, for ground fields totally real anything other than $\mathbb Q$.) First, whatever description one chooses for "Eisenstein series" (attached to cusps?), a suitable weighted average of the level-7 (for example) such should be level-one. A literal notion of holomorphy tells us there is no level-one, weight-two such. Thus, there are (at most) six linearly independent Eisenstein series at that level, so not quite possible to "attach" one to each cusp. It is not hard to say more.
Then there is the further issue of expressing various Eisenstein series in terms of each other, by the group action. At square-free level, the underlying (!) representation theory is simpler (Iwahori-fixed vectors in principal series are well understood, at least up to a very useful point.)
But, at this point, without knowing more precisely what the questioner wants, or may discover is wanted, there are too many things that can be said to know which to choose to say. :)