Some of these techniques will take a stain and some will not.
If you have a wood chip or gouge on a piece of furniture that won’t be readily visible you can use carpenter’s glue or a cyanoacryate like Mister Glue’s Instant Glue. The end product will not take a stain so best to use this type of fix for areas that will be painted or not visible. This can be done a couple of different ways.
If you have a very thin chip or area to fill you can put in some sawdust and then a few drops of Instant Glue which will wick in and make for a very hard filler. Sand it smooth and you are done. This will set up very quickly. Another way of doing this is to use white or yellow carpenter’s glue.
Make a fairly dry mix. Use it to fill a bottom tear out, a void in plywood, a chip or a gouge. Let it dry then sand smooth. The wood glue may stain a bit but the Cyanoacrylate will not. I’ve had many people use the CA (cyanoacrylate) to do repairs in wood carving as well but remember that it won’t stain so use it judiciously.
If you are not putting stain on the finished product you can often, with practice, get the sawdust-glue filler to match the color of the wood – as long as you are using the same wood, obviously.