The Fair Housing Amendments Act (FHAA) requires most landlords to modify their pet policies to permit emotional support animals even if pets aren't generally allowed. It does not apply to ALL landlords, but does apply to most. If your landlord owns more than four units and is not a church, then he is probably subject to the FHAA.
We have complete instructions on how to request a modification of "no pet" policies over on the main site at:
http://servicedogcentral.org/content/node/138Emotional support animals are not required to be specially trained, other than the ordinary training pet dogs receive to make them pleasant to live around. In other words, they should be well housebroken and not disturb neighbors with nuisance barking.
As Shidash explained, an emotional support animal is different from a service dog in that because it is not specially trained, the owner has no particular right to take it in public accommodations other than those which ordinarily permit pets. It typically takes 18-24 months to train a service dog.
An emotional support animal (ESA) need not be a dog. It could be a cat, bird, snake, almost any species which is permitted for pets. Some landlords can impose a restriction on what species or size emotional support animal you get, if you are already a resident before acquiring the ESA. If you have the ESA before moving in, they usually can't put a restriction on size or species.
If you live in section 8 housing, you may have some additional rights and responsibilities.