The former, the number of contracts in one specific position.
Let me cite an example of too many. Say you buy call options on XYZ stock whose open interest is currently, at 250. You decide to buy 200 of it at say $2.00. Now, assuming it is a thinly, traded stock option, you have most of the contracts available. Now, the stock for some reason drops like a rock and now you have huge losses and wish to exit. Problem is nobody wants to buy your call option back at the $2.00 or higher price. Now, it is being bid at $0.20 by the market marker and he only wants 100 of it. You are now stuck and cannot get out of your position. Now, you can sell the 100 contracts at $0.20 each but, you still have 100 contracts that you will not be able to sell.