Forums are community wide message boards. They're usually divided up into topics ("General", "Music", "News and Politics", etc.) and people are free to start what is called a new post or "thread." People can also reply other poeple's posts/threads.
I have to say, SpaceHey forums are not ideal forums and lack a LOT of the key features of most of them. There's not a way to know if someone has replied to you, there's no way to subscribe to threads to follow them, and you can't reply directly to individual people. The layout also buries real posts amongst a lot of spam, and only new threads show up at the top of the board. I hope they continue to improve the forums for the future because forums are a hugely fun and engaging space to meet people, get questions answered, and to generally chat and have fun.
Blogs are personal and individual. Consider a blog a public diary or journal entry. The Bulletins appear to be more like quickfire life updates, fun things to share, or little blurbs you don't mind seeing vanish after a week's time.
Groups are little online clubs you can join that have their own private forums for the group members to discuss whatever the group topic is about.
Also welcome to SpaceHey!