It sounds like the disconnecter system may not be engaging.
The disconnecter system is on the left side of the trigger group.
There is a silver-colored lever pointing forward, that fits ABOVE a blued lever pointing to the rear.
Check these levers for free movement and proper assembly.
If the disconnecter has somehow gotten out of position, or is fouled, the hammer won't drop when the trigger is pulled.
It's very possible the gun is dirty and gummy, or there may be foreign matter jamming the action.
Take the trigger group out of the receiver.
Spray it dripping wet with Rem-Oil, using the spray to blow any fouling out of the assembly.
Allow to drain.
Shake off the excess lube, and test the action.
To test the disconnecter assembly:
With the group out of the gun, put the safety OFF.
Hold your finger in front of the hammer so it CAN'T fall all the way.
If you allow the hammer to drop all the way, the trigger group can be damaged.
Pull the trigger.
The hammer should release.
Hold the trigger to the rear, and re-cock the hammer.
The hammer should be held by the disconnecter.
Release the trigger and the hammer should return to position.
Pull the trigger.
The hammer should drop again.
If the assembly isn't working properly, take the group to a gunsmith.
It's possible to disassemble the trigger assembly and replace parts, BUT NOT IF YOU DON'T KNOW HOW.
If you MUST work on it yourself, buy a copy of Jerry Kunhausen's book "The Remington Shotguns" A Shop Manual".
This is available from Brownell's, Midway, and most book sellers.
This gives full directions on how to work on the Remington's.
Remington parts are available from Brownell's and Gun Parts Corporation.
Best guess is: Either the disconnecter is out of position, or the trigger group is fouled and dirty.
You very rarely see actual broken parts in an 870.