Requirements for the computer vary considerably, depending on the piano software used and the choices you make in software settings. Just as with digital pianos, sample-based software is highly dependent on the size of the computer's memory, while physical modeling software — which creates the sound in real time rather than retrieving an existing sound sample — primarily depends on the speed of the computer's processor. At a minimum, hardware requirements will involve processor type and speed, and the amount of random-access memory (RAM) and hard-disk space. These requirements range from packages that can run on most recent-vintage mid-range computers, to those requiring higher-speed processors, 4 Gigabytes (GB) of RAM, over 250 GB of free hard-disk space (preferably on a 7200rpm drive), and a dedicated sound card. Either way, you need to check the hardware requirements of the individual software package you'd like to run to make sure it will work properly on your computer — or use it as an excuse to get a new computer.
Aside from making sure that you have enough memory to store and run these packages, processor and sound-card choices will also keep latency in check. Latency is how long it takes the computer to produce a sound from the time you press a key. When latency becomes noticeable, your brain doesn't know whether to slow your playing so that the sound can catch up, or to speed up to make the sound happen faster. Neither of these works. (Anyone who plays the pipe organ knows what latency is, and will adapt to it without a second thought.)
This is where the real fun starts. There are currently over two dozen software-piano packages available, at prices ranging from $79 to $895. These include both sample-based packages and packages based on physical modeling. Several host acoustic pianos (i.e., the sources of the samples) are available via software, including instruments made by Bechstein, Bösendorfer, Blüthner, Fazioli, Kawai, Steingraeber, Steinway, and Yamaha. If you'd like to add some period instruments to your palette, there are also packages with samples from historical fortepianos.
If you're not particularly into computers, software pianos may not be for you. But if you enjoy even a mild bit of tinkering, and have dreamed of owning a collection of the world's finest pianos or even of "designing" your own piano, you may find software pianos an irresistible temptation. If you're interested in following the world of piano software, it's discussed in Piano World's "Digital Pianos — Synths & Keyboards" forum.