What kind of video card can I put in this motherboard for HDMI?  A PCI card?

You have a few alternatives..the black port is"agp graphics port" ,the Advanced Graphics Port..you also have a PCI-E port which is the longer of the 2 PCI ports..that port is faster (graphics wise)than the others you got..the smaller white one(at the bottom),is normal PCI Port..(the slowest of all for graphics..but to answer your question...buy a "HDMI to VGA cable"..tha way you will be able to use the onboard graphics of the motherboad..(saving resoures.,of your computer-you only have a limited number of resoures on any computer..).

Search for "Graphics Card PCI-E x16 HDMI".

looks like PCI-E, but hard to tell with that cable in the way.

what model is the computer?

Your photo's a bit blurry but the brown slot appears to be a PCI Express x16 slot, which is what almost every modern graphics card has.  It may be an earlier, slower version of it but they're backwards compatible so that's not a problem.  Almost every card you can buy will fit that slot, but you need to look at the sticker on the power supply to see how much power (watts) it can put out.  You either need to choose a card that uses a low enough amount of power for your power supply to handle, or buy any graphics card with a PCI Express x16 slot that you like and change the power supply for one that can run it.  Don't forget to make sure that the card you choose also has the HDMI output that you're after.

PCI-E Graphics card

Should bring the computer to a local computer repair store and they will do it for you.