Hilbert presented only ten of the problems (marked with a *) in his actual talk; however all 23 appeared when the list was published in Göttinger Nachrichten in 1900.

This list is quoted from Constance Reid's biography of David Hilbert. My own comments appear in italics.

Notice that Fermat's Last Theorem does not appear in the list, although problem 10 might be considered a generalization of it. Hilbert did mention the problem in his introduction, however.

1. *Cantor's problem of the cardinal number of the continuum, and whether the continuum can be well-ordered.
The Continuum Hypothesis has since been proven undecidable, and Ernst Zermelo showed that the well-ordering of the continuum is equivalent to the axiom of choice.
2. *The compatibility of the arithmetical axioms.
Is mathematics consistent? Godel's incompleteness theorem shows that this cannot be proven. Fortunately for Hilbert, he died before Gödel produced his result.
3. The equality of the volumes of two tetrahedra of equal bases and equal altitudes.
Hilbert's description is a bit misleading, (read this for a better explanation) as the problem relates to the equidecomposibility of the tetrahadra. Max Dehn proved this to not be true in 1902.
4. The problem of the straight line as the shortest distance between two points.
Solved by G. Hamel
5. Lie's concept of a continuous group of transformations without the assumption of the differentiability of the functions defining the group.
Solved for compact groups in 1933 by John Von Neumann, and generally in 1952 by Deane Montgomery and Leo Zippin working together, as well as Andrew M. Gleason working independently from them.
6. *The mathematical treatment of the axioms of physics.
A can of worms? We now have the Standard Model, GUTs, superstring theory and M-Theory. All have problems.
7. *The irrationality and the transcendence of certain numbers.
Most importantly, Hilbert wanted a proof that 2sqrt(2) was transcendent. A.O. Gelfond proved that 2sqrt(-2) was transcendent in 1929; given this Hilbert's student Carl Ludwig Siegel quickly proved the transcenence of 2sqrt(2).
8. *Problems of prime numbers.
Most importantly, the crucial Riemann Hypothesis, to which so many other things have been linked, has yet to be proven. Neither has the Goldbach Conjecture.
9. The proof of the most general law of reciprocity in any number field.
10. The determination of the solvability of a Diophantine equation.
Movement on this problem had to wait until Alan Turing introduced computability theory. The problem is considered to have been "solved" in 1970 by Yuri Matiyasevich, building on work by Martin Davis, Julia Robinson, and Hilary Putnam over the preceding two decades. As late as 1999, Mr. Matiyasevich considered that there was work yet to be done. ariels recommends the book Hilbert's 10th Problem for those wishing to pursue the subject further.
11. The problem of quadratic forms with any algebraic numerical coefficients.
12. The extension of Kronecker's theorem of Abelian fields to any algebraic realm of rationality.
13. *The proof of the impossibility of the solution of the general equation of the 7th degree by means of functions of only two arguments.
14. The proof of the finiteness of certain complete systems of functions.
15. A rigorous foundation of Schubert's Enumerative calculus.
16. *The problem of the topology of algebraic curves and surfaces
17. .
18. The expression of definite forms by squares.
19. The building up of space from congruent polyhedra.
20. *The determination of whether the solutions of "regular" problems in the calculus of variations are necessarily analytic.
21. The general problem of boundary values.
22. *The proof of the existence of linear differential equations having a prescribed monodromic group.
23. *Uniformization of analytic relations by means of automorphic functions.
24. The further development of the methods of the calculus of variations.

Disclaimer: I am not a professional mathematician and cannot pretend that I understand what all of these problems are or even their significance. I do claim understanding of some of them. It needed to be noded, that's all I can say.