A good way to tackle this issue is to check the dictionary definitions and example sentences (Oxford has good ones, and also fraze.it). Then try to explain the differences to yourself and then post your own example sentences for correction and feedback.

"Intellectual" means involved or interested in activities that are mostly to do with the mind, such as mathematics, cultural criticism, art and so on — mostly abstract things. "Intelligent" means clever, or mentally able. Traditionally it means good at abstract reasoning, but these days we acknowledge that there are different kinds of intelligence, such as emotional intelligence or even physical intelligence. "Intelligible" means able to be understood. For instance, an intelligent intellectual might say something indistinctly, mumbling, and so they wouldn't be intelligible (they'd be unintelligible). Or, they might say something so complicated that I couldn't understand it; again, unintelligible. But then if they said it again more simply, then they'd be intelligible. "Smart" is a less formal word for intelligent, like clever. Smart, clever and intelligent each have slightly different meanings in different places. Clever, for instance, sometimes has a slightly bad implication, like "sneaky", but this varies from place to place so don't worry about it too much — any of them will be understood.