The black market is a place for players to buy units that have been produced by various independant manufacturers across the Inner Sphere. It's accessible via the black market tab.
Most units on the black market are created automatically, and in most cases they are specifically sold to the black market. If a faction's hangars get too full, however, they can also ship units off to be sold on the black market. This process is random, and it becomes more likely as more units of a particular type accumulate in faction hangars.
Units are listed for sale for 48 ticks, or 12 RL hours. The bidding process is a silent auction, where players may make bids(at a cost of 20 influence each), and the high bidder at the end of the auction receives the unit. Money is not withdrawn until an auction is won, so be careful not to spend too many C-Bills if you have active bids outstanding. In order to see the unit being sold, you can click the Show Unit button. If you change your mind about a bid, you can retract it and guarantee that the money will not be spent, but you will not get the influence back.
There is an option in the menus when right-clicking a unit to sell it on the black market, but this functionality is disabled on the server at present. Nothing will happen if you try to sell a unit.