I could be wrong, but I am not sure if there is a rule, per se, could be named. But I think the simple matter of number agreement is applicable here. That is, plural nouns should be matched with plural articles.
It is simply this: "lots" is plural; "lot" is singular. So, whenever you use "lots" you do not use the indefinite article because it is always singular. "lot" is inherently singular, and must take the indefinite article. And the form of the verb must also agree as to number.
Correct (pairs with identical meanings):
- Lots of mail keeps arriving.
- A lot of mail keeps arriving.
- There are lots of ways to skin a cat.
- There is a lot of ways to skin a cat.
- A fat lot of good it did me!
- Lots of good it did me!
Note carefully that the latter two examples are both negatory, using sarcasm to convey: I didn't get a lot of good out of it! Oddly enough, if I took the last one and said "It did me a lot of good" unless your tone of voice suggested otherwise (or the context suggested otherwise) it would be taken to me that it did indeed do you a lot of good. But that's a digression, sorry.
Lots of lots!
The word "lot" in the context of your question means a non-definite grouping or number of something or other. There might be only one or there might be multiple such groupings. It's a wonderfully vague term, which means it has great utility. There is however a couple of more restrictive ways to use it.
One is a specific bounded amount of land, which is a "lot". Sometimes combined with other words to indicate the use of the lot, e.g. "feedlot", a piece of land whereon cattle are fed. In this sense of the word, if I said "I have lots of land" it would be a double entendre -- it would have a double meaning.
Another sense of lot is the outcome of a chance event. "To throw lots" means to toss dice or a coin to allow chance to choose an outcome. "It is your lot to be a poor man" means that fate or circumstance has determined that your condition in life is to be poor. The word "lottery" comes from this root meaning.