I expect everyone's worked out that tweets look different if your mouse is hovering over them, but if not, compare (unhovered-over tweet at top) and contrast (same tweet beneath it, but this time hovered-over).
Because it's my tweet I get the option to delete it - highlighted by the red oval. All tweets have a star option (yellow highlight) that appears when you hover over them, clicking on this allows the tweet to be favourited. Incidentally you can find out which of your own tweets have been favourited by using services like Favotter (http://favotter.matope.com/en/) or Favstar (http://favstar.fm/).
Notice the bit, which I've not circled, that says 'about 4 hours ago from web' - there can be up to two links in this phrase (though in this particular example there's just one).
Were you to click on the 'about 4 hours ago' bit you'd go to a page which contains the tweet by itself, and has this URL http://twitter.com/JoBrodie/status/7237369855. This can be useful if you want to direct someone to a particular tweet, rather than to someone's general stream of tweets.
Where it says 'from web' it might equally say 'from Twitterfeed' or 'from Tweetdeck' and clicking on these will take you to the company's link telling you about these services. 'From web' doesn't take you anywhere.
If someone's replying to someone it may say 'in reply to XYZ'. Clicking on that will usually take you to the tweet to which they're replying - useful for following a thread backwards - but only if they used the 'Reply' function (note that the 'in reply to XYZ' link doesn't distinguish between a tweet that's sent by clicking 'Reply' and a tweet that someone starts typing, putting the person's @name first).
The pic below needs to be read in reverse... Tweet 3 is the first in a series of tweet exchanges that I posted to my Twitterstream, Tweet 2 is @Zeno001's reply to me, and Tweet 1 is my reply to him. If you saw these tweets in your Twitterstream (you could view these if you were following both of us, or if you clicked on either of our profiles) then you could follow the thread. Admittedly it's not the most fascinating exchange but I think the gist of how to follow a thread is clear.
If you use a service like Tweetdeck then when you 'reply' to someone then it's possible that the threading of tweets will be lost (it depends on whether or not the end reader is viewing on the web, or via phone I think).
I think the unusual thing here is that there are 'hidden' links on a webpage - this seems to me to be something that disappeared fairly early on, on most websites (perhaps an accessibility issue - it's now usually very clear what is or isn't a link).
