Streaming usually refers to the process of watching TV/movie/video clip etc online in your web browser. The programme may be a live stream e.g. watching a football match online, or pre-recorded e.g. watching a movie online or listening to an internet radio station.
Usually there is no specific download required, since its viewed in the browser. Examples of streaming sites are RTE player (used for watching various programmes screened on RTE in the last couple of weeks, BBC iPlayer (similar to RTE player, but not sure if that works in Ireland), MyP2p.eu (very handy for watching most sporing events), YouTube etc.
Yes its legal. Be aware that streaming video is quite heavy on bandwidth so you should be mindful of your BB monthly download limit. If using mobile BB then the speed may not be fast enough to stream the media and you may need to wait for it to buffer it before watching - fine if watching a rerun of Fair City, not so good if you want to watch a match in real time