What does Francis Underwood's ring mean to him and why does he bury it?

The ring probably symbolizes a simpler time for Frank. As we saw in his class reunion episode (Season 1 Episode 8), Frank was quite different and more idealistic back in school. He sang in a barbershop quartet, was in a loving relationship, and basically ran wild and rambunctious with his schoolmates. The fact that for the duration of his stay in his school he not once broke the fourth wall to share his plans and cynical world view is very telling. It wasn't until the very end of the episode, when Frank has left his school and friends behind, did he start talking to the audience again.

At the civil war site, he discovered that an ancestor of his was buried in the memorial. This probably inspires him to bury his class ring there as well, and with it his idealistic past. By this point of time, Frank was well on his way in fulfilling his plan to rise to power, and already has blood on his hands. It probably didn't seem proper to keep wearing that reminder to his more innocent self.

When Frank finally reaches his ultimate goal, a new ring is forged for him. Not the ring of the innocent and idealistic Frank, but a ring for the ruthless power hungry Frank who will unflinchingly stomp on his enemies and all that come in his way.

There is an awesome answer, on Reddit, here.

At one point, Frank explains to Tusk why he taps his ring, when he leaves a table or a lectern. To quote from the Reddit posting by thisisntnamman:

Frank: "Something my father taught me. It's meant to harden your knuckles so you don't break them if you get into a fight. It also has the added benefit of knocking on wood. My father believed that success is a mixture of preparation and luck. Tapping the table kills both birds with one stone."

To me, Frank was only tapping his ring, not his knuckles directly. He was only symbolically following this father. He wasn't fully preparing himself for the fight. By burying his ring he will have to hit his hands directly and toughen them up for the big fight with Tusk, over who will control the president and the future.

I didn't recall this interchange but I went back and verified that Frank does, indeed, say that to Tusk

My reading is similar (but not the same) as G Blake Meike's.

The ring symbolises a more noble/idealist Frank, taking it off isn't to allow him to toughen his knuckles, but is him "taking the gloves off", and entering the bare knuckle fight.

The dialogue with Walker is a a Freudian thing, his humanity (or conscience) warning Walker that the fight is now on as he has taken the gloves off.

Burying it at the civil war site, is Frank inspired by his ancestor to fight whatever the cost to him, but realising that the battle there was of more value/more just than his, it is a place to bury his humanity before what he must do.

The replacement ring comes as Frank has won the fight, and can now put the gloves back on, crucially it is a new ring as it no longer signifies the nobility of Frank, it's now a token to remind him of what he has done to get there.

I believe he buried that ring when he learned that his ancestor fought on the losing side. He buried that ring because he was dissappointed, because he thought he is the descendant of winners, not losers. He buried it because he didn't want to be a loser, but to win the trials that awaits him.

When he received the new ring, it didn't symbolize the loser, but the winner, because now it had a new meaning: a token of power.