These are the backing boards that I use -- each has it's specific application and there are pros/cons for each but all of the backings I use are weather proof.
WEDI BOARD - This is what I use for nearly all my mosaics. I go to Phoenix to buy it (it's cheaper there) but is still very pricy compared to the other products. It has a cement layer with steel cables that sandwiches a foam core and is very strong and rigid -- a 3x5' board weighs about 7 pounds. I usually cut it into quite irregular shapes with a small electric jig saw but it can also be cut easily with a utility knife. I use 1/4" thickness in frames and for smaller mosaics and 1/2" thick for larger items. A big advantage to Wedi is that it is very light weight, so finished mosaics are easy to hang using either a copper wire or a 'wedi hanger' that I purchase from mosaicsmalti.com.
EASY BOARD is about 1/4 the cost of Wedi and is easily cut into any shape with a utility knife or jig saw. It is available from Home Depot or Lowe's and is very light weight ( maybe 6 lbs. for a 3x5' sheet). It is pretty flexible and 'bendy' so I don't use it very often -- only in small frames where it is well supported. But if I were doing mosaic with large pieces that had it's own rigor, I would consider using it more. It is important to seal the cut edges when using easyboard or it won't be waterproof.
HARDIBACKER, aka Sheet Rock is a cement board that is available at Home Depot/Lowe's and is also about 1/4 the cost of Wedi. Unfortunately, it is so heavy that I can barely lift a sheet -- a 3x5' sheet of 1/4" hardibacker weighs about 40 pounds. And you have to store it laying flat (I used to put it on my garage floor and straddle my car over it) or it warps badly. If you score it with a utility knife, it is possible to break it by hand but it breaks only in square or rectangle shapes. I've heard of people cutting it with a saw into odd shapes but the dust from that is a serious concern -- even when you break it it is necessary to wear a mask to avoid the dust particles. If I were making square tables that I wanted to be heavy, I would use this product but the weight of it makes it difficult to use on hanging products unless it is permanently screwed into the walls and tiled over.
ROCKRITE is a cementous board that comes in 2x4' boards and is used mainly in countertops because it doesn't require plywood underlay. It comes in a 2" thick board that can be scored with a utility knife and broken. It's a good product to use on a bench top because it is strong and rigid. I'm thinking of making some sculptural pieces with this product -- will let you see pics when I get around to doing that!
DENSHIELD is a gypsum board that is also used for tiles. It comes in a 4' square sheet that is inexpensive. It cuts easily with a utility knife or jig saw -- but it's big disadvantage is that it is absolutely necessary to wear a mask and the dust is the kind of dust that you can't clean up easily...it sticks onto everything and stays there. I tried this board but ended up giving it away!
WOOD -- when would I use wood? What about that fabulous old table that you want to put tiles on? If at all possible, I would securely attach wedi board to the table top before tiling it. Or if the wood were thick enough, I might try to have a 1/2" deep area routed out so I could sink wedi into it -- then do the tiles on top. I know there are more modern processes where wood is treated to be very dense and coated like for kitchen cabinets -- most of those would be fine to use for mosaics. At least I wouldn't hesitate to use them.
So, the most important point to take from all this is that when you are making your mosaic, the backing board matters a great deal. Don't hesitate to give me a call if you're struggling to decide what to use for backing and I'm glad to help you think through your options. Just remember, the effort you're putting into your creation and the expense of the tiles you are using really justifies thinking this issue through carefully. Have fun!