Slackware Linux is a GNU/Linux distribution which is developed by Patrick Volkerding. In contrast to many other distributions it adhers to the so-called KISS (Keep It Simple Stupid) principle, which means there are no graphical tools to configure the system. In the beginning configuring the system can be somewhat difficult, but more experienced users love the transparancy and flexibility this approach provides.
Another distinguishing feature of Slackware Linux also complies with the KISS principle: Slackware does not have a full-blown package manager like RPM. Slackware packages are normal tgz (tar/gzip) files with an additional installation script. Tgz is much more powerful for experienced users than RPM and avoids dependency problems many package managers, like RPM, have. Another well-known Slackware feature are the BSD-like initialisation scripts. Slackware has one script for every runlevel/task instead of bunch of scripts which are symlinked in different runlevels. This allows you to easily tune your system without having to write new scripts yourself.
The learning curve of Slackware is a bit steeper than most other distributions and you have to be prepared to invest some time. Once you start to understand the distribution you will probably find it much easier to tune than many other distributions. It is comparable to cooking: a microwave dinner is very easy to cook. You put it in the microwave, wait a few minutes and it is finished. Cooking without a microwave requires more discipline, you have to get acqainted with ingredients and cooking techniques. But once you have learned the art of cooking it is easy to cook things which taste much better than a microwave dinners.
There are several ways to get Slackware. First of all you can download Slackware Linux from the internet using a FTP mirror. A list of mirrors can be found on the Slackware website, http://www.slackware.com/. Although Slackware can be obtained freely it is not a bad idea to buy the official CD set. Patrick Volkerding invests a lot of time in the development of Slackware Linux, by buying an official CD set you are supporting Slackware development.
That depends on what you want to use Slackware Linux for. A 486 with 16MB RAM is just enough to run XFree86 with a lightweight window manager like BlackBox or Windowmaker and light X applications. Such a machine is also sufficient for making a simple webserver or FTP server. A machine with even less memory (e.g. 8MB) can still be used as a router or firewall. Heavy-weight desktop environments like KDE and GNOME require a faster computer, at least a Pentium with 32MB RAM, but you will probably need 64 MB RAM or more to do something useful. The advantage of Linux is that Unix-like applications such as vi, gcc and apache run quite fast on older machines. Most other distributions also have that software, but have heavy installation and configuration tools which make running Linux on an older computer a pain. Running the newest Slackware on a 386 is not recommended, though some FTP mirrors contain older versions of Slackware (even back to 1994) which work quite well on a 386.
Not in the same sense as Linux From Scratch or Gentoo, you do not have to compile the whole system. Proponents of such distributions often believe in huge speed increases as a result of e.g. CPU specific optimizations. In practice the speed increase is very small, you might even not notice it, with the exception of a few applications (like MPEG-2 decoders and maybe desktop environments such as KDE). Slackware is normally precompiled, but if you want to you can still compile parts of Slackware yourself using the Slackware sources and build scripts. The advantage of Slackware compared with source distributions is that you do not have to compile the system, which provides much more flexibility and probably a more stable system (because some optimizations break things).
Nobody except Patrick Volkerding knows. Many people monitor the Slackware-current Changelog, it provides good information about the development of Slackware Linux.
Install the "mysql" package from the "ap" diskset. First you have to set up the default database. You can do this by executing mysql_install_db. After that you need to set the permissions for the directories in /var/lib/mysql by executing chown -R mysql.mysql /var/lib/mysql/*. Now you can start the MySQL daemon with safe_mysqld. Do not forget to set your root password with /usr/bin/mysqladmin -u root password 'new-password'.