Beach bingo refers to a game of bingo put on by a for profit business. There is a fee to play the game, usually charged per-card, and the payouts are small.
In North Carolina the game of bingo is considered gambling, and therefore illegal. But North Carolina has a very long tradition of playing bingo, particularly in the coastal and outer bank areas, so it wasn't made 100% illegal. There are two* exceptions to the bingo ban.
1. Bingo is legal if held by a charitable organization that has a current Charitable Bingo License.
2. Bingo is legal if there are no prizes offered that are worth more than $10. This second category is both popularly and legally referred to as 'beach bingo', presumably because it gained its popularity in the coastal areas of NC.
Beach Bingo cards usually cost about $10 (3 for $20!). Prizes may be cash or items, but no item may be worth more than $10. Often there will be multiple winners per round (the first five people to get bingo all win, for example). Because the prizes are so low, it benefits the parlor to have a high win rate, encouraging people to buy more cards. It is usually allowable for a person to play multiple cards at the same time, although you may be allowed to collect only one prize per round.
Beach bingo is so popular in Eastern NC that is is often taken as a synonym for bingo. It is stereotypically the pastime of old people and families on summer vacations at the beach. Tourists may be somewhat disappointed to find that large signs proclaiming "Up to $8,000 given away each night!!!" refer to the number of prizes, and not the size.
* If you live in NC you may have been to a bingo game put on by your business, college, etc. that belongs in a third category; those that charge no admission. In this case there is no limit to the amount that can be given away, and no permit is needed (but IANAL).