Alot
Alot is a misspelling of a lot. It's not considered a word and should not be used in formal writing.
A lot
A lot means a large number or amount.
A lot is a collocation—any combination of two or more words that go together so consistently that they can be treated like one word, e.g. nuclear disarmament.
It can take up two different grammatical roles in English:
(1) A lot can act as a quantifier, such as in a lot of water, or a lot of people, similarly to phrases like a cup of coffee, or a crowd of people.
In this case, technically, it is a noun phrase, as it appears to be, like a cup (of . . .), a box (of . . .), a bunch (of . . .), etc., where originally, the word lot meant “a share of something” or “a portion of something” and it can still be used this way in old-fashioned English. Lot also has other old-fashioned meanings related to “share,” referring to a kind of vote, or to a person’s fate. Just remember that a lot of . . . works exactly like a bunch of . . . .
A lot used as a quantifier: Don’t spend a lot of time worrying about word origins!
(2) A lot can also be used as an adverb, with the same meaning as the quantifier:
A lot as an adverb: We like you a lot. Do you want to hang out a lot with us?
Notice that this is used similarly to adverbs like awhile and apart. But a lot remains a lot, not alot! There is no logical reason for this inconsistency in English spelling. Some phrases change more slowly than others. A while changed into awhile over time, but a lot has yet to change.
Allot
Allot means to give someone or something a share of something or to divide something into shares.
Each speaker is allotted fifteen minutes to speak.
The school allots the classrooms based on course enrollment.
While a lot can act as a quantifier or an adverb, allot is simply a verb (transitive).