The belief I have on dreams is that theres different types of dreams. These are to name a few, with some characteristics.
Messages: Message dreams are normally messages from ether your higher self (your soul), a deity, another person, or a spirit (like a passed on loved one). They can be very direct, but a lot of times its more of a code. Someone is trying to tell you something basically. It may be something you need to do or need to know, information about whats happening in a situation, a warning, or just something you need to hear or someone needs to tell you.
Past Life: Dreams that are memories of a life you lived before. These are usually very vivid, feel very real in every sense, and feel familiar even though you don't know the places or people you're with.
Projection: Projection is when your mind leaves your body and goes elsewhere, ether around the current universe you're in, a alternate universe, or another realm. Within the current universe, everythings the same but you physically aren't there. It would be described as being a ghost or being invisible. You may be able to walk around, go other places of the world, watch people, etc. Alternate universes are sister universes which are similiar to our world, with versions of the places here and the people here but different choices were made there than were made here, which makes that universe different than this. A place may look differently because someone decided to paint the walls yellow instead of blue, a person might be different because they decided to work at a book store rather than a gas station. Another realm is basically another world, but many things about it are different or don't exist here. These dreams are also usually very vivid and confusing.
Psychic Dreams: These are dreams of things that are yet to happen, most people already know what this is so I don't really need to explain it I suppose. Only thing I would say is that they don't always happen because the future has a way of changing, even with a small choice that was made.
Some dreams are just dreams, just parts of the imagination that are running wild while your body's asleep. They could be emotionally triggered, or memories, etc. Not all dreams really have a meaning.
When I have a dream, I try to figure out the type of dream and then go from there. As for the dream you had about your cat, I can relate to that, I lost my cat I had for sixteen years, I still dream about her and I miss her very much. Personally, I believe the dream is probably a message type, ether a message from your cat - just telling you that they're there for you in spirit, or its a emotional trigger, you miss your cat and its being expressed through your dreams.