"Your bed is near the door."
Translation:Твоя кровать около двери.
What is the difference between all these 'near' words? Около, возле, у
Yes, it is because of the preposition около. It is, however, the genitive case.
"Около and возле are equivalent prepositions that require the genitive case and are close to the English preposition 'near' - "около здания" means "near the building", and so does "возле здания".
At least, that is what dictionaries tell you; from personal experience, when describing city planning около has a connotation of a more broad vicinity than возле. It may have to do with the etymology: Proto-Slavic *kolo ("circle") + the prefix o- ("around") would imply "somewhere around that place".
Близко is very similar in meaning and in syntax requirements to the English 'close' - it is an adverb, and as such it requires another preposition, like от. However, to me "близко от здания" sounds a little off, so I'd use the equivalent synonym недалеко in this context - "недалеко от здания".
Близко along with недалеко and рядом are used in standalone sentences like "Где театр? - Он близко/недалеко/рядом." ("Where is the theatre? - It's close/not far from here/nearby.")
Около and возле are also adverbs as well as prepositions, and can be used in a standalone sentence ("Я живу тут, возле" - "I live here, nearby"), however it may sound a little strange, especially with около."