Those are standard PCI slots and that card will indeed work with them.
However, if you are intending on installing Windows 7, you're going to have some serious problems with speed or even installing on such a very outdated system. Since video cards were last on PCI, AGP has come along and now PCI-express x16 version 2.1 (about to be 3). If you want to upgrade to Windows 7, I very strongly suggest getting a new computer altogether. If you only intend on just browsing the web (no gaming beyond simple card games or watching videos beyond SD), then you don't need to upgrade. PCI for video cards is so outdated. PCI-express x16 version 2.1 is today's standard (about to be version 3).