How do I replace a P Trap that is glued to the wall pipe?

A pic here in the age of instant pics and selfies would be helpful. But... start with the removal of the crap from the ABS (black) or metal drain coming horizontally from the wall. Use pliers, heat gun, 220 sandpaper, whatever it takes to get that ABS or metal pipe clean. Then rebuilding is just replacing everything else. And it's less than $20.

There are slip-on connectors that will slide over the cut end of pipe, once you have cut the damaged end of the wall tube off.  You can use those connectors to hook up a new wall tube, which will connect to the new P-trap (assuming you're replacing it).

Get the repair coupling, P-trap and replacement wall tube first, before you do any cutting.  And take measurements of existing pipe diameter.  Being residential yours is probably 1-1/4 but double check anyway.  When you do cut the wall tube, err on the side of not cutting off enough - you can always cut off more if necessary.

Good luck with it.

First, make sketch of the pieces and how they fit together. Then disconnect the piping and lay it out end to end for what needs to be replaced and how it ties back in to the good piping. Take a photo of the tie-in. Now, take it all to the hardware store and get some help and get all the new parts to rebuild it. Especially the gaskets and nuts and the tie-in to the old pipe. Take the new stuff home. Make the tie-in to othe old pipel and rebuild back to make the complete replacement. It could be that you might have to cut a piece of old pipe. And you will have to make that cut in the right place depending upon the new part you will buy that will make the connection. Don't cut it unless you have the new part to make the connection.

Where is this pipe? Bathroom? Under kitchen sink?  Anyway, I have never heard of a pipe being glued into the receiving pipe. Have you tried twisting the P-trap pipe to see if the outlet end really is glued into the wall pipe? Another strategy - can you cut off the P-trap pipe a fe inches from the wall to attach a new fitting to insert/attach the new P-trap pipe and secure that pipe. {Then the top of the new P-trap pipe gets attached to the downspout of the sink drain that is above it.}  Good luck. [[ For a temporary "fix" to use the sink put a bucket under the down drain, and (as needed) dump the sink water down the bath tub drain or the toilet. ]]