What is the difference/similarity between "take a long time" and "take a while"?
This process takes a long time.
This process takes a while.
Can I use take time without using a in a sentence?
- This process takes time.
- This process takes a long time.
- This process takes a while.
All of the above are correct. You can't say "takes long time", though. When time is a mass noun, like in the first sentence, you can drop the article: "this process takes time" (or "this process takes money" or "this process takes guts"). When you say "a long time", you are basically using time as a count noun, which means you need the article.
You can read up on count nouns on Wikipedia. The relevant piece is reproduced here:
The concept of a "mass noun" is a grammatical concept and is not based on the innate nature of the object that the noun refers to. For example, "seven chairs" and "some furniture" could refer to exactly the same objects, with "seven chairs" referring to them as a collection of individual objects but with "some furniture" referring to them as a single undifferentiated unit. However, some abstract phenomena like "fun" and "hope" have properties which make it difficult to refer to them with a count noun.
Classifiers are sometimes used as count nouns preceding mass nouns, in order to redirect the speaker's focus away from the mass nature. For example, "There's some furniture in the room" can be restated, with a change of focus, to "There are some pieces of furniture in the room"; and "let's have some fun" can be refocused as "Let's have a bit of fun".
Replace "fun" with "time" in those last examples and you have the same idea.