What size wire for my 240 v 50 amp sub panel
Get a nice BIG panel. Panel spaces are cheap when you're buying the panel, and 50A@240V will support more stuff than you think!
You need to bring over a 4-wire feed.  2 hots + neutral + ground. Must keep neutral and ground separated at the subpanel.
At only 125' you do not need to upsize the wires for voltage drop.
You cannot use NM cable.  If you use UF cable (25&quot; burial depth) you need #6 copper but I don't recommend copper on feeders this large, waste of money.
Any other type of cable or wire, you'll need #8 copper (again not recommended) or #6 aluminum or larger.
Whatever you pick for conductors, ground must be -2 of that (e.g. if you use #2 aluminum then #4 ground).  This is because of your 50A objective.
Direct burial is 24&quot; cover above the top of the wires.
Conduit in the ground is 18&quot; cover above top of conduit.
Rigid Metal Conduit or Intermediate Metal Conduit is 6&quot; cover, except under vehicle pathways it must be 12&quot;.