I had a look at Paulo Ribenboim's "13 lectures on Fermat's last Theorem" (Springer Verlag, 1979). In section 3 of Chapter III, he discusses (with full bibliographical details) the controversy around Euler's proof, and then provides a proof, using purely elementary number theory, which he attributes to Euler.
This is an interesting question that I've wondered about myself, so I can't really answer it properly but I'll make a couple elementary observations. First, for a lattice in ${\mathbb Z}[\tau]\subset{\mathbb C}$ to be an ideal just means that multiplication by $\tau$ takes the lattice to itself. For example, for the Gaussian integers this says the lattice is invariant under 90 degree rotation about the origin. It is easy to see that this means the lattice is a square lattice with a basis consisting of two of its shortest vectors. (This is really just a disguised version of the Euclidean algorithm for Gaussian integers.) Thus every ideal is principal in this case.
Back to the general case, another observation is that if two ideals are in the same ideal class, then the two lattices are related by an orientation-preserving similarity, that is, rotation and rescaling, which is what multiplying an ideal by an element of ${\mathbb Z}[\tau]$ does to a lattice. (It seems the converse should be true as well). For example in the case $\tau =\sqrt5i$ the principal ideal class consists of rectangular lattices similar to ${\mathbb Z}[\tau]$ itself, and the other ideal class (the class number is 2 here) consists of lattices similar to the lattice $(2,1+\sqrt5i)$ which is skewed rather than rectangular. It is enlightening to draw a picture to see how this lattice is invariant under multiplication by $\sqrt5i$.
Another thing that can be viewed geometrically is the correspondence between ideal classes and equivalence classes of binary quadratic forms of fixed discriminant. An ideal, viewed as a lattice, determines a quadratic form by restricting the usual norm (squared) $x^2 + y^2$ to the lattice, then renormalizing suitably to make equivalent ideals have equivalent quadratic forms. A textbook that explains this, to some extent at least, is Advanced Number Theory by Harvey Cohn.
It would be interesting to work out some more examples to see what the different similarity classes of lattice-ideals look like, especially in cases when the class number is larger than 2. Is the structure of the ideal class group somehow visible in how the similarity classes are related?
Best Answer
In his article On the "gap'' in a theorem of Heegner, Stark does a pretty thorough job of explaining where people thought the purported gap came from, to what extent it actually was a gap, and what you would need to fix such a thing if it existed. I'm paraphrasing, but he basically argues that the confusion stemmed from some errors (typos?) in some analytic results of Weber that Heegner had heavily used. So in a literal sense, Heegner had not proved it because he had cited faulty results, but Stark shows that he deserved credit for the theorem since using Heegner's argument with the correct versions of Weber results (which were indeed known to Weber), the job gets done.
Here's the mathscinet review of the article:
http://www.ams.org/mathscinet-getitem?mr=241384