That's a (front) sway bar end link. They are relatively inexpensive and easy to DIY replace, except that if the nut or sleeve is rusted on then you might need a hacksaw or angle grinder to cut it off, and with both sides the same age I would go ahead and replace both.
You can probably find a low grade version (stock was also low grade so don't dismiss them too quickly) at Autozone/etc at around $15 each, that have soft riding rubber bushings and carry a lifetime warranty.
Elsewhere like Rock Auto, you can get the same thing for less even after shipping cost, or get an upgraded part with polyurethane bushings, but little or no warranty. The polyurethane bushings are firmer and age better.
Either way make sure not to over-tighten the nuts, IIRC they need somewhere around 15 ft lbs (could be off a bit), don't highly compress them or they'll blow out the bushings prematurely.
They are easier to replace with either both sides jacked up or vehicle left on the ground so you aren't fighting against the tension of the sway bar.
Your sway bar bushings (one is lower left in the first pic) themselves are starting to look a little crusty and cracked too. I might go ahead and replace them while you're doing the job, as a replacement pair is usually around $5 to $15 at Rock Auto. Again there may the choice of rubber vs polyurethane but practically everyone goes with polyurethane.
In some cases if the end links are rusted on, it can be easier to take the whole sway bar off first, get it out away from the vehicle to cut the end links the rest of the way off. If you get sway bar bushings, measure your bar diameter to be sure you get the right size. Maybe on an '08 they're all the same but I know on older generations the diameter could vary.
If you do the main sway bar bushings, you might want to sand rust off the sway bar where it makes contact and use a good amount of silicone grease lathered on the bar for lubrication or else you may get squeaking later. The rubber has less potential to squeak without it, but I'd still use grease if you have some.
If you don't want to buy a big expensive tube of grease, some "might" come with grease or you can use the cheap little packets of silicone brake grease sold at auto parts stores - It will be clear to milky white, just silicone, some PTFE in it wouldn't hurt but not the brake grease that's another formulation with copper or ceramic, etc in it.
The end links do not need grease but if you have a rust issue in your region, they might rust less if you grease the metal portions but try not to get grease on the bolt end where the nut sits or it might loosen itself over time.
Having recently done this job (again) on my older Explorer, I ordered the Moog Problem Solver end link set (MOOG K700542, the one with white bushings) which has been popular among Explorer owners, but recently they switched to an inferior half hollow bushing design so I would recommend against getting those. Energy Suspension probably (not sure about your model year) makes one of the better sets as their bushings are among the best, but probably costs more too.
On the above link, the ones costing around $3 (each, not a pair) are the same as those at Autozone for $15.