In the case of an elliptic curve over a global field of positive characteristic (i.e. over the function field of a curve over a finite field), Tate reinterpreted the BSD conjecture in a more geometric way.
Namely, an elliptic curve over the function field of a curve $C$ over a finite
field $\mathbb F$ can be "spread out" to form an elliptic surface $S$ over $C$ (i.e. a surface mapping to $C$ whose generic fibre is an elliptic curve).
Giving rational points on the original elliptic curve corresponds to giving sections of the projection $S \to C$. To determine such sections essentially amounts to determining all the curves lying on $S$ that can be defined over $\mathbb F$.
Now to determine these curves, one can look at the cycle class map which takes
any curve to its class in the second etale cohomology group of $S$ over $\overline{\mathbb F}$. Since one is considering curves defined over $\mathbb F$, the image of this map lies in the Frobenius invariants of the etale $H^2$,
and Tate showed that (the rank part of) BSD is equivalent to the statement
that every Frobenius invariant element actually arises from a curve defined over $\mathbb F$. (He was then led to make his general conjecture, known as
the Tate conjecture, which I have discussed here.)
There is a general philosophy, known as Iwasawa theory, which tries to take
intuition from the Weil conjectures and the Tate conjecture (which are about
varieties over finite fields) to formulate analogous statements for varieties over number fields.
The idea is that passage from $\mathbb F$ to $\overline{\mathbb F}$ should be
replaced by passage from $\mathbb Q$ to $\mathbb Q(\zeta_{p^{\infty}})$ (i.e. adjoin all the $p$-power roots of $1$, for some prime $p$). At least
if $p$ is odd, the Galois group of $\mathbb Q(\zeta_{p^{\infty}})$ over $\mathbb Q$ is pro-cyclic, just as $Gal(\overline{\mathbb F}/\mathbb F)$ is.
Unlike in the finite field context (where one has the Frobenius element), it does not admit a canonical generator, but we can just choose a generator; traditionally it is labelled $\gamma$.
Now if $E$ is an elliptic curve over $\mathbb Q$, one can construct a certain Galois cohomology group attached to $E$ over $\mathbb Q(\zeta_{p^{\infty}})$, which will have an action of $\gamma$ on it, which is analogous to the second etale cohomology group of the elliptic surface in the function field case.
It is the action of $\gamma$ on this Galois cohomlogy group that Coates is referring to.
In fact, the Galois cohomology group in question is the Selmer group of $E$
over $\mathbb Q(\zeta_{p^{\infty}})$, and the main conjecture of Iwasawa theory for $E$ over $\mathbb Q$ relates the characteristic polynomial (actually, characteristic power series, but let me not get into that detail here) of $\gamma$ on this Selmer group to the $p$-adic $L$-function of $E$.
There are various caveats (e.g. as I'm describing it here, the conjecture only makes sense if $E$ is ordinary at $p$), but let me say that in broad terms, the two (characteristic power series and $p$-adic $L$-function) are supposed to be equal up to multipication by a unit in the ring of power series; that one divisibility was proved by Kato; and that more recently the other divisibility (and hence the main conjecture itself) was proved by Skinner and Urban.
Knowing the main conjecture does not actually imply BSD, since it relates the Selmer group to the $p$-adic $L$-function (rather than the usual $L$-function), and since it doesn't deal with the problem of proving that Sha is finite. But it is natural to make a $p$-adic BSD conjecture, and the main conjecture is closely related to this. Unfortunately, it doesn't actually imply $p$-adic BSD either (even if one grants the finiteness of Sha), because of possible non-semisimplicity of the action of $\gamma$ on the Selmer group. (This echoes the problem, in etale cohomology, of proving that Frobenius acts semisimply --- an important problem that is open in most situations.) Thus $p$-adic BSD is also currently open (as far as I know).
Finally, although the main conjecture is weaker than $p$-adic BSD, which is in turn different to the usual BSD, there are relations between all three, and in particular, with the main conjecture proved, the only obstruction to the following statement:
- the $L$-function of $E$ vanishes at $s = 1$ iff the Mordell--Weil group of $E$ is infinite
is the finiteness of Sha. (I.e., if we could prove that Sha is finite, we would
get the preceding statement.)
For a (much more technical) discussion of the main conjecture and $p$-adic BSD, one could look at the text of Colmez's Bourbaki seminar.
Your phrasing of the problem, in terms of two ways to compute the rank (analytic and algebraic), is not quite correct. Rather the conjecture states that two numbers are equal.
On the one hand there is the rank of the elliptic curve, which is as you said: the number of independent rational points of infinite order. In the context of BSD, one calls this "the algebraic rank", $r_{alg}$, of the elliptic curve $E$.
Then, there is another quantity attached to $E$, also a non-negative integer, but of quite a different nature. To describe it, we begin by defining the $L$-function of $E$,
as a certain Euler product (a Dirichlet series given by taking a product
indexed by the primes), denoted $L(E,s)$, which is holomorphic in the half-plane $\Re s > 3/2$. Thanks to the modularity results of Wiles et. al. we know that this function has analytic continuation to the whole complex plane, and so in particular, we can consider its order of vanishing at $s = 1$. In the context of BSD, we call this "the analytic rank", $r_{an}$, of $E$. (Note though that this is just a name; $r_{an}$ is not defined in terms of the rank of any abelian group, but rather is the order of vanishing at $s = 1$ of an entire function; the only reason for calling it a "rank" comes from the BSD conjecture.)
The BSD conjecture is then that $r_{an} = r_{alg}$; in other words, we can determine the rank of $E$ by determining the order of vanishing of $L(E,s)$ at $s = 1$. To my mind the most striking consequence of this is that
$r_{an} > 0$ if and only if $r_{alg} > 0$; in other words, we (conjecturally) can determine whether or not $E$ has infinitely many rational points (i.e. has positive rank) by evaluating $L(E,1)$ and determining whether or not it equals zero.
In fact one direction of this weaker form of the conjecture is actually known: it is known that if $L(E,1) \neq 0,$ then $E$ has only finitely many rational points; this is due to Gross, Zagier, and Kolyvagin, with another proof more recently by Kato.
(I won't give the definition of $L(E,s)$ here; you can find it in many places,
including in Wiles's write-up I'm sure, or just by googling "L-function of elliptic curve".)
Best Answer
Here are my two cents: some references 1 and 2.