What is the modern axiomatization of (Euclidean) plane geometry? I have heard anecdotally that Euclid's Elements was an unsatisfactory development of geometry, because it was not rigorous, and that this spurred other people (including Hilbert) to create their own sets of axioms.

I have two related questions:

1) What is the modern axiomatization of plane geometry? For example, when mathematicians speak of a point, a line, or a triangle, what does this mean formally?

My guess would be that one could simply put everything in terms of coordinates in R^2, but then it seems to be hard to carry out usual similarity and congruence arguments. For example, the proof of SAS congruence would be quite messy. Euclid's arguments are all "synthetic", and it seems hard to carry such arguments out in an analytic framework. 

2) What problems exist with Euclid's elements? Why are the axioms unsatisfactory? Where does Euclid commit errors in his reasoning? I've read that the logical gaps in the Elements are so large one could drive a truck through them, but I cannot see such gaps myself.
I can recommend an article Old and New Results in the Foundations of Elementary Plane Euclidean and Non-Euclidean Geometries by Marvin Jay Greenberg, The American Mathematical Monthly, Volume 117, Number 3, March 2010, pages 198-219. One of the great strengths of the article is that I am in it. Marvin promotes what he calls Aristotle's axiom, which rules out planes over arbitrary non-Archimedean fields without leaving the synthetic framework. If you email me I can send you a pdf. 

EDIT: Alright, Marvin won an award for the article, which can be downloaded from the award announcement page GREENBERG. The award page, by itself, gives a pretty good response to the original question about the status of Euclid in the modern world. 

As far as book length, there are the fourth edition of Marvin's book, Euclidean and Non-Euclidean Geometries, also Geometry: Euclid and Beyond by Robin Hartshorne. Hartshorne, in particular, takes a synthetic approach throughout, has a separate index showing where each proposition of Euclid appears, and so on. 

Hilbert's book is available in English, Foundations of Geometry. He laid out a system but left it to others to fill in the details, notably Bachmann and Pejas. The high point of Hilbert is the "field of ends" in non-Euclidean geometry, wherein a hyperbolic plane gives rise to an ordered field $F$ defined purely by the axioms, and in turn the plane is isomorphic to, say, a Poincare disk model or upper half plane model in $F^{\; 2}.$  Perhaps this will be persuasive: from Hartshorne, 


  Recall that an end is an equivalence class of limiting parallel rays


Addition and multiplication of ends are defined entirely by geometric constructions; no animals are harmed and no numbers are used. In what amounts to an upper half plane model, what becomes the horizontal axis is isomorphic to the field of ends. This accords with our experience in the ordinary upper half plane, where geodesics are either vertical lines or semicircles with center on the horizontal axis. In particular, infinitely many geodesics "meet" at any given  point on the horizontal axis.
Any axiomitization of 2-dimensional Euclieadn geometry is unjustifiable. In such a system, you just assume without proof that the undefined concept of distance satisfies certain intuitive properties and prove other properties from it like the Pythagorean theorem. If you invent a space and explicitly define a lot of the relations on it like distance and show that those definitions satisfy the intuitive properties of 2-dimensional Euclidean geometry, it follows that that space can be represented by R^2 and that for any points (x1, y1) and (x2, y2), the distance from (x1, y1) to (x2, y2) is sqrt((x2 - x1)^2 + (y2 - y1)^2). We can define the distance formula to be sqrt((x2 - x1)^2 + (y2 - y1)^2) without giving a reason then show that it satisfies the intuitive properties of distance. After we show that it satisfies the intuitive properties, we can derive from that fact that that the formula is sqrt((x2 - x1)^2 + (y2 - y1)^2) but there's no need because that can be proven directly from the definition of distance. Some people might want to know why it was defined that way so for them, it might suffice to show that that formula is the unique formula that satisfies the intuitive properties of distance. Those people might also be satisfied with seeing the distance formula being proven by defining it them showing that it satisfies those intuitive properties then reproving it from those properties because from that, they can figure out how to prove that that formula is the unique formula that satisfies the intuitive properties of distance. The article http://speedydeletion.wikia.com/wiki/Distance that I wrote actually gives an incomplete proof that that formula is the unique formula that satisfies the intuitive properties of distance. A complete proof probably includes constructing the set of all real numbers with operations from the power set of the set of all natural numbers in Zermelo-Fraenkel set theory then showing that that set is a totally ordered field, similar to what I described in my answer at What is a natural number?.