I've heard the phrase, "We gotta ball game". It could also be "We have a ball game". But I don't understand the meaning of "having a ball game". If anyone has heard this expression before, please help. I'm so very confused after guessing about this for a long time. The person who said it couldn't even tell me what he meant by this.
The prototypical phrase is "We've got ourselves a whole new ball game"; often meaning that the situation has so changed that one has a sporting chance of success in whatever the situation is. Without the "whole new" part, it means that regardless of whether the situation has changed, one has a chance of success.
Idiomquest.com says, regarding "whole new ball game":
When the state of affairs change dramatically and presents new opportunities.
Edit: As others have mentioned, a phrase like we have a ball game may mean that a contest has developed, in a game that formerly looked like a walkover. That might be what was meant by the person who said it, and of course is the sort of thing meant by the phrase a sporting chance in my answer above.
Looking at ngrams, it appears that the phrase new ball game is found a hundred times as frequently as any of got a ball game, us a ball game, or have a ball game; and looking at links from ngrams, the contest-has-developed sense is surprisingly rare, but of course does occur.
American sportscasters and sports fans will often say we have a ball game (often preceded by now or suddenly) when two sports teams seem equally matched or when the score of a game gets close, thus avoiding a blowout. A blowout could be seen as the opposite of a ball game in this sense. Another way of thinking about it is that having a ball game is synonymous with having a competitive matchup that is fair and interesting to watch.
This phrase has been borrowed from sports jargon and is commonly applied to other competitions, such as a close political race.
At @Fumble's request, here's some more examples of the phrase used idiomatically:
When "we DON'T have a ball game," it suggests that the "match" is one-sided. "It's a ball game again," or "we've got a ball game" means that a recent event "suddenly" made a formerly lopsided game competitive.
Say one team is leading another in an American baseball game 5-0. In professional games, at least, that represents almost a sure win for the first team. But suppose the occurrence of the unlikely event that the trailing team hits a "grand slam" home run (for four points). Now, it's 5-4, not 5-0, and the formerly one-sided game is suddenly "competitive" (especially since the trailing team is still "at bat," and might add more runs).