I have used MatLab...they tried to teach me how to use it last semester at college...they didn't really succeed :)))
What Do You Use?

I use Mathematica, but I don't really like it. I have not used Maple and haven't seen Alpha. I have also used MatLab a lot, but I was recently introduced to python by one of my professors. It's open source, and very powerful. It can be set up to mimic MatLab (on Windows and Linux anyway) using a GUI called Spyder, or it can mimic Mathematica using a program called Sage. Given the cost of Mathematica/MatLab/Maple, I would definitely point you in that direction. It's a little bit of a pain though, because you have to install a lot of packages to make it useful (such as Matplotlib and numpy and scipy.)

They're computer programs that let you calculate more advanced math stuff. They're not always introduced formally as I understand, but sometimes they're in courses.

Does a slide rule count?
Just kidding, never used a slide rule seriously, though my cousin did. I did have one and sometimes was able to use it, but not very often. I had a programmable TI that was slow enough it blinked the LEDs to show you it was still working on the problem and not hung in a loop. Still beat the hell out of calculating square roots by hand. I wonder if they still teach that...
Are these newfangled programs very costly? My daughter will be into the fun stuff in a few years, wondering if I need to start a savings program.

I have used Maple and Matlab quite a bit, and when I was an undergrad the math department at my school used both. There were some professors who preferred Maple and some that preferred Matlab. Seems like it depends on what you want to do. I think Maple is easier to learn how to use, it is basically like a really big calculator with all kinds of cool functions, but it is not as good as Matlab in setting up more programming related stuff (if-then statements, for loops, etc.). I really liked Maple when I used it, mostly because of the ease of use, but I suspect that Matlab is a bit more powerful (depending on which packages you get with it; things like simulink are super powerful, but that also means that it often takes more time to learn. Also there are a ton of add on packages to matlab to make it behave like many other programs, I think there is even one to emulate Maple). Maple was never as good dealing with matrices as Matlab, but if I had a fairly simple math problem (crazy integration, differential equation, simplification, solve for, etc.) it was always much easier to punch it into Maple than matlab and get an answer much quicker. Also Maple writes out equations in a really pretty format (as opposed to simple text, which can make some equations difficult to read in Matlab).
Mathematica is also super powerful, but I don't have much experience with it. I do know that it is very popular, but I have also heard that it can be a little harder to learn than Maple. I believe that Wolfrom Alpha works in much the same way as Mathematica, so that should give you an idea of what the program is like.
There are also other packages such as Mathcad, but most people think it is not as good as mathematica (although some people have said that it is easier for simple stuff).
Bottom line was that I usually felt that I could use a combination of Maple and Excel to solve most problems, but there was that occasional oddball problem that I needed Matlab for.
As far as purchasing any of these programs, if it is for school, make sure that is what they are using, because you will usually have to use what they dictate rather than what you prefer.

I'm a musician. I use Pro Tools. While good at math, I only took it as far as was mecessary for degrees. The extent of my math program skills is greating a decent "graphing calculator" in MS Excel.

What's wrong with pen and paper? (Or a black board and chalks if you're doing crazy stuff)
I just use Python, which comes with a good maths module, but Mathematica seems really interesting because of all the built-in functions it has.

I wonder if they still teach that...
Are these newfangled programs very costly? My daughter will be into the fun stuff in a few years, wondering if I need to start a savings program.
They do in some places! I actually just use Java or C++ when I need some major number crunching power. Unless it is Matrices... Then I have a copy of Matlab which I got at a Math competition a while back... And for proof-writing, I use TEX.

I've used maple a bit, but only for computing derivatives and solving a couple of differential equation. I have used MatLab extensively in my masters degree to do bifurcation diagrams and other stuff like that.
If you need to do this sort of stuff, and your system is not too non-linear, I would recommend trying to use MATCONT which is a GUI for MatLab, that has all sorts of nice features.
Hello, I was curious as to what my fellow team mates use in terms of mathematics programs: Have you used Maple? Have you used Mathematica? Have you seen Wolfram Alpha?