Summarize:

"A lot" is a noun phrase (an indefinite article a + a noun) that functions as an adjective to refer to the quantity of something. This makes it like any other noun phrase, such as "a wagon" or "a puppy," and it is always spelled as two words. For example: I have a lot of money. I have a lot of burritos. I have a lot of Bratz dolls. I like you a lot. "A lot" is often used in informal conversation to describe the quantity of something, but it is usually considered too colloquial for use in formal prose. Some people prefer to use a lot to describe quantities for collective nouns, or things that cannot be counted individually, such as sugar or water. For example: "There was a lot of sugar on the floor." It may be preferable to use other words, such as "several," "many," or "a great deal of" when referring to count nouns (things that can be counted individually). Allot is a verb, something you do, rather than an adjective that describes something. For example: He was allotted three feet of space to set up his art stall. I will allot these lottery tickets to my friends. It no longer exists as an acceptable word in English. It was taught in the U.S. school systems in the 60's through the 90's which may be causing some confusion between generations. If you catch yourself writing alot, even in informal situations, rewrite it as a lot as it is the accepted appropriate use.
Use the informal phrase "a lot" when you wish to describe quantity. Understand that "a lot" is often considered slang. Use "allot" when you are referring to giving out, apportioning, dividing or granting something. Refrain from using "alot". Practice your usage.