When we got an oil change, he said it's "leaking, but not out of fluid or low". Do we need to do any replacement/maintenance on it, or just wait until it gets worse? How would I check something like this?
With most vehicles of the rear wheel or four wheel type this would invole crawling under the vehicle or raising it onto service ramps/jackstands.The plug would be removed and an object inserted in the hole (a little finger works well for this)to see if the differential fluid is even with the bottom of the plug hole,if oil gets on your finger the level is OK.If the leak is at the rear cover or the pinion(where the drive shaft attaches) monitor your driveway or parking space for drips that would indicate it is getting worse.It appears you have found a mechanic that is thorough enough to have noticed the leak so ask him/her to check it while having your oil changed.If the leak is at an axle seal it is something you will want to have repaired sooner as it could impact brake performance if the oil reaches the brake linings.
If it runs out of fluid it will need to be replaced, because it will destroy itself rapidly (probably preceded by a loud grinding noise). In the meantime, as long as there is sufficient gear oil to wet the gears it will be fine.
As for severity, it is common for differentials to develop a "seep" around the front seal where the drive-shaft connects, or the gasket where the two halves come together. If this is just a "wet spot" on the differential, without generating actual drips of oil, then it is not a big deal - just something to keep an eye on to make sure it doesn't get worse (actual drops on the ground).
If it is just a "seep", or even a very slow leak (where drips are generated, but very slowly), then short/medium term, you can just check the fluid and top it off periodically. On an old vehicle, this might be a fine solution - just check it whenever you change the engine oil. On some vehicles, it can be very difficult to check the fluid, which makes this a less attractive proposition.
The "correct" repair would be to have the differential overhauled, which would involve removal from the vehicle, opening the differential, installing new wear-components like bearings and thrust washers, then reassembling with new gaskets and seals. Parts cost would be in the $100-$300 range, with labor in the 4-8 hour range. I would not bother with this unless the leak seems to be getting worse, or is regularly leaving drops on your driveway.