AMD and NVidia usually provide a reference layout for their graphics cards that many manufacturers use. If a card uses this reference layout there is essentially no difference between cards from different manufacturers.
But the manufacturers often try to differentiate their graphics cards from those of the competition, so they use different cooling solutions or overclock the cards a little. Those cards will vary in GPU temperature and noise mainly, the overclocked cards will be a little faster. You'll have to research those differences for each card. If you value a silent PC you should research carefully, there can be sigificant noise differences between cards.
The 9800GT also has an Eco version that uses less power and can be powered from the PCI-Express power alone, it does not need an additional power cable from the power supply.
Your mainboard has a x16 PCIe slot if I'm not mistaken, any PCIe card will fit in there. You'll need a power supply that can deliver enough power for the card you choose (the GPU alone uses a bit less than 75W for the Eco card, a bit more for the standard 9800GT). Any reasonably-built computer should be able to power a 9800GT, but if you have a very crappy power supply you should be careful.
The 1GB 9800GT has more memory than the usual 512MB for that card. If you're not gaming at very high resolutions this does not help you with most games. There are some exceptions, especially if you're playing with modded high resolution textures for some games. But if there is a significant price difference, I would recommend the 512MB version.
As Jeff pointed out, the connectors can vary between cards. You should check that the card has the proper connectors for your monitor.
This is a VGA connector (D-sub 15), which is an analog connection. Only use this one if your monitor does not have any other connectors available.
This is a DVI connector (image from Wikimedia/hartmut.krummrei), it is a digital connection, but it can also transmit an analog signal. If your graphics card has only this connector, you (almost certainly) can use a DVI-to-VGA adapter that is usually supplied with the card to connect it to a VGA-only monitor.
There are more connection types, but you won't usually find them on older, cheap monitors.
If you don't need a very quiet card, just take the cheapest one you can find.