You don't need a dedicated server or specialist operating system to run MySQL, or any of the PC optimised server engines. However depeneidng on you application (requirements) it may help. For most small oraganisiations the db can run happily on a file server or even standard PC. As your data gets parger and the demands placed on the server get larger then moving it to a dedicated machine may make sense as you can tune that machine to mget the best out of the db.
There is nothing intrinsically different about PC based servers and standard workstations (with the exception of how the machine is configured, servers often have built in redundancy (to allow the server to keep working if say the power fails, of a disk drive dies), use higher grade componenets (eg gigabit network rather than 100Mbs).
The use of memory also may very - on a non dedicated PC you may allocate less memory to the database
So to recap if you want to go down the MySQL option then all you need to do is download the server (I'd recommend you also download the MySQL Administrator & Query Browser at the same time , and Workbench assuming it gets finished). all these are available form MySQL. THe only choice is whihc version to choose - in my view its ano brainer, if you have a free choice go version 5, it answers many of the competitors claimed weaknesses in previous versions of MySQL, with things like stored procedures, triggers etc....
After that it depends on what development environment you use to access the data, you may need to set up ODBC data sources on the client PC's. But thats no different to other server engines.
If you want to make life easier in the short to medium term you could consider using MS SQL server, I think they have released a freeware version to offset MySQL's march into what Microsoft see as 'their' territory - it may be better for your CV and mdedium to long term employmnet prospects.
I'd rather be riding on the Tiger 800 or the Norton