Microsoft SQL Server is a commercial product and mySQL is free, though there is a free express edition of sql server 2005. The two programs have very different roots.
Microsoft SQL Server originated in the short-lived collaboration between Microsoft and Sybase, which had a SQL Server product that was very popular on UNIX machines in the financial world. The SQL dialect of SQL Server, T-SQL is still very much the way Sybase designed it, but Microsoft's product has been rewritten/overhauled several times. mySQL is the product developed by MySQL AB from Sweden which was founded by David Axmark.
mySQL is popular in LAMP web sites (Linux, Apache, Perl, mySQL) built with no-cost products. You will not find many mission critical databases such as airline reservation systems run on mySQL. mySQL is, just as many open source products (mySQL is free but not open source) supported by an active user community, though it is possible to get support from MYSQL AB. Microsoft's SQL Server is supported by Microsoft.
SQL Server is a much more powerful product with many enterprise features that make it a good choice for large databases. There is also excellent programming support available, from a client perspective (ADO, ADO .NET, OLEDB) and inside the SQL server (with SQL 2005 you can write programs in managed .NET code that execute within the server). But often, all that is needed is a simple and low-cost or free product, and that's where mySQL is fine.