From its initial implementation in Minecraft Alpha v1.0.6 until Alpha v1.0.11, cactus behaved slightly differently from how it works today.
Cactus still generates in clusters consisting of cacti that are up to three blocks tall, however, they will not grow any taller, meaning the only way to get more cactus is to go exploring and find more. When you first click on the cactus to break it, you will take half a heart of damage.
Alpha cactus takes up an entire block, and can be placed adjacent to a block without breaking. There are also no limitations for what blocks cactus can be placed on. You do not take damage if you walk along the edge of a cactus block, damage is only done if you walk on top of the block.