The big stack is a poker term meaning the player with the most chips at a poker table. This term is usually reserved for a player with significantly more chips than any other player. It is also sometimes called being chip leader at a table. (e.g. "You're big stack now! You can't go broke!")
A large stack of chips itself is also sometimes called a big stack (e.g. "You've got a big stack!"), or a deep stack. This is true even if the stack is not the largest stack at the table - in cash games some players define any stack larger than the maximum buy-in is a big stack. There can be more than one player with a big stack, but only one player is the big stack.
The opposite of big stack is short stack.
Deep stack table
In no limit games, a deep stack table is a table with larger minimum and maximum buy-ins than are typical for the stakes involved. The typical minimum buy-in is generally reckoned to be 20-25 big blinds, while the maximum is 100 big blinds. By contrast, buy-ins at deep stack tables will typcally range between 50 and 200 big blinds. Many online poker rooms offer specially marked deep stack tables, while a growing number of brick-and-mortar rooms are also increasing their buy-ins to these levels.