(1) Lothar Gottsche's program counting curves of various genera on various rational surfaces. (It was used to create his remarkable conjecture.) Depending on your browser, clicking on these programs will show the code, and shift-clicking will let you download them.
Caveat from Lothar: "People can use it, and also distribute it if they wish, as long as nobody holds me responsible for eventual errors, and if they manage to figure out without any documentation how to run it."
The program as originally sent to me by Lothar.
The program with some documentation added by me. (This is probably the version you should take.)
In this version, I've added a bit to help compute invariants of del Pezzo surfaces, to get examples for my paper "Counting curves on rational surfaces".
(2) Carel Faber's wonderful program to compute intersections on the moduli space of pointed curves. New version posted Feb. 10 2004.
Here is the file Numbers.tar. "tar -xvf Numbers.tar" creates the directory "Numbers". Read the file "Instructions", which mainly consists of examples.
Caveat from Carel: This version appears to work for Maple 9, Maple 8, and Maple 7. In general, the condition 3g-3+n less than 20 is required. Naturally, no guarantees can be given, but on the other hand, no errors have come to light since 1998.
(3) Anders Buch's useful Littlewood-Richardson calculator is available here. (Anders has moved to Aarhus, so at some point this link will be dead; I'll ask him for a new one.)
(4) I've written a maple program "galois" to check whether my criterion for the galois (monodromy) group of a Schubert problem is at least alternating (and incidentally calculating the enumerative answer) is here. It's not documented, so you may want to ask me for useful commands. For example, "read galois; biggalois(3,7);" will check *all* Schubert problems on G(3,7).
(5) I've also written a program working out my flag Littlewood-Richardson rule in (everything up to) five-space. Available upon request. (It is no longer continually being updated.)
(6) Jim Bryan has written a program essentially working out the entire Gromov-Witten theory of a curve. In other words, if you want to compute anything, including local Gromov-Witten invariants, you can use this. It is available here
(7) Dung Nguyen has written a program implementing all the formulas in his remarkable papers computing the characteristic number of rational curves, nodal rational curves, cuspidal rational curves, elliptic curves with fixed j-invariant, and elliptic curves with fixed complex structure in projective r-space (r up to 5 --- this condition might later be removed). The program is here. The documentation is here. The source code is in this directory.
Other interesting programs: Andreas Gathmann has written a program calculating genus 0 descendent invariants. Andrew Kresch has written a program FARSTA calculating quantum cohomology.