The two phrases are so similar that the average english speaker would most likely not notice if you use them incorrectly.
However this is how it works:
“lot” becomes the subject of the sentence
“lot’s of” is a plural(p) lot
“a lot of” is a singular(s) lot
therefore our “to be” verb must match the quantity for “lot”
Examples:
1.) There are(p.) lots(p.) of dogs
or
2.) There’s(s.) a lot(s.) of dogs
(There is —> There’s)
Without markings:
1.)There are lots of dogs
2.)There’s a lot of dogs
When speaking english, if you were to always use “There’s” nobody would ever notice this is wrong when combined with the plural lots. Only when speaking though, it would be wrong to type “There’s lots of dogs” on an important paper or such.
There is one other case for this:
If the mentioned noun is countable or not
In this case we will use “milk” which is usually uncountable
(milk can be countable if you have multiple different *types of milks*)
When using an uncountable milk it works as such:
1.)“There’s(s.) a lot(s.) of milk(s.)”
or
2.)“There’s(s.) lots(p.) of milk(s.)”
Without markings:
1.)There’s a lot of milk
2.)There’s lots of milk
So, as you can see, our “to be” verb remains singular because our milk is uncountable
Overall, when SPEAKING english, if you always use “There’s”, your sentence will ALWAYS sound correct and you can use either “lots of” and “a lot of” as you wish.
But when writing formally, that is how it works to my knowledge
Sorry this explanation is so long, it’s such a small difference it’s difficult to explain in short!
Hope this helps!