I damaged a small component (doesn't appear to be a chip) on a video card (it's not working) and I was wondering if I could replace it.

Ooo, another one of these.

Your photos aren't very clear, but it doesn't look like any of the surrounding components are marked, so the most direct way to guess what got destroyed is not possible. Unless you can find the exact same model of card, lift the component you damaged, measure it, and source something spec'd correctly for the circuit you're taking it from (voltage/power rating, possibly more depending on the circuit), your chance of successfully repairing this card is pretty low.

Pro-tip: don't use power tools on things you aren't prepared to destroyed.

Can't see much in those pictures.

It's a very small straw to grasp - but a lot of circuitry on such boards can be repetitive. So you could look to see whether there is a similar pattern of component layout, elsewhere on the board. There may be several, or none. If you can find a pattern the same, and the still present components are the same - then there is a fair chance that the missing components will match their cousins.

A long shot - but million to one chances do happen, nine times out of ten.

This board is going to be multi-layer. Not just what you see on the top and bottom, but a sandwich of several such layers. I can't see how deep that centre point hole is - but it may have broken a track on an intermediate layer. Back to million to one odds again.

Although others may shoot me down in flames over this - capacitors often aren't that critical when it comes to value. In some cases, even whether they are present or not. I have got things working in the past by guessing and trial and error. With capacitors.

Resistors,on the other hand,tend to be far more fussy. As they often alter the dc biasing of a semiconductor, on which its basic operation tends to hang - they can be kill or cure.

The good news is that it only hit passive components. So a bit of investigation elsewhere on the board might be worth the time. Doing that to a millipede chip that Murphy will insist is a custom job not otherwise obtainable - and it's out with the hot air gun. Salvaging components on the board that might live another day, elsewhere.

I’ve done a lot of soldering for various projects I’ve built - and I’d suggest you chalk this one up to experience and just get a new card. It will always be at the back of your mind that the soldering is sketchy and the replacement parts a guess at best. Worst case - fire and/or toasting your mobo and components down the line. Best case - do the repair and throw it in a junker system and stress the crap out of it before leaving it in a plugged-in-left-on machine.

The resistor looks fine, but you nicked the nearby trace (the slightly raised brownish line). It's hard to tell from your blurry photos (make sure you steady your hands/camera and have good lighting when taking closeups) but it looks like the broken trace was running to the resistor pad. The fix would be to scrape the soldermask off of the trace to expose the copper right before the break, and then use a piece of fine (e.g. 30 gauge) wire to jump from the exposed copper to the pad.

If you don't have a good fine-tip, temperature-controlled soldering iron, flux, fine-tipped tweezers, and lots of practice using all of the above, you might want to try to find someone who does, or at least do a bunch of practicing on a sacrificial board first.

edit: Looking at your photo again it looks like there might have been a third resistor in between the two that were still there. At that size there are no identifying marks on them, so without either a schematic/layout, or an undamaged board that you could remove the resistor from and test, you are probably out of luck. Also, at a second look that damaged trace could just as likely be going to a via near the resistor pad and not to the resistor—again really hard to tell without an undamaged reference board.

Photo is blurry, but it looks like the area you identified (not near the mounting hole) was partially drilled into after the bit snapped off? The distance between the top and first inner layer of a PCB is usually only 5mil or so. If there was any signal in the way on any layer drilled through, all bets are off. I can even see one of the top traces is severed. You'd need to reconnect that and replace the components with similar values to have any chance of success.

I agree with other people that cap values aren't usually that critical. But an EE could approximate what the function of the cap was and get a close value. You may not be able to do that. Resistors are more finicky, but even that might not be true. If it's a pull up, then you'd have a wide margin.

Clean off each affected pad, reconnect any broken traces by scraping the soldermask and soldering a jumper, then approximate and reconnect components. Identify and make sure VCC and GND aren't shorted, and you'll have a 50/50 shot at fixing it, and a reasonable chance at not destroying your motherboard. But no guarantees there :)