Every profile name is preceded with @. If I start a tweet with @ZimloidSpacer, you'll see it as an interaction. You can then click "reply" and write something back. These are all public, but they only show up directly to people who follow all concerned parties - other people see them when they visit your profile, for example. There are also direct messages, which are private, and you can only send direct messages to people who follow you.
# is how you start a tag. You can make your own tags - it can be anything you want - no spaces or special characters, perhaps except a few. They are clickable and show other tweets containing the same tag. So they're useful when you're talking about something other people are talking about or might be interested in. For example, you could add #DoctorWho when you tweet your paintings, so that people who search for #DoctorWho (there are a few
) find them. But you could also have some fun with the tags. #RidiculouslyLongWordThatNooneCanPronounce is one of my favourites.
Trends is what other people talk about. It is quite interesting sometimes. Often, it makes me realise how little I know about pop culture, for example. It's useful, though.