This post assumes you're using Twitter via its desktop / web browser version (http://twitter.com) but you can do some of these things on smartphones (presumably tablets too but I've never used one). Twitter is just a tiny part though as people will post information from your tweet on Facebook (or just share content that you've posted directly there) so this shouldn't be taken as covering everything.
Also, as always with these blog posts I'm assuming that there's knowledge and skill I've not discovered yet and if you have it I hope you'll share it in the comments or ping me on Twitter, ta :)
Summary for those who don't need any explanation or detail:
1. Click on tweet to expand, count RTs
2. Search for @mentions of the tweet sender, to pick up manual RTs
4. If tweet contains URL search for and count those
A thing that is quite helpful to know
Every tweet has its own web address which can be found in its timestamp (which says things like '4h' if recently tweeted or '10am, 5 April 2013' if more time has elapsed), or in the details link once you've expanded the tweet - the page has lots of information about the tweet including how many people have retweeted it and who they are, and other tweets threaded in conversation with that tweet.
1. Click on tweet to expand, count RTs
Click anywhere on anyone's tweet to expand it and count the number of times it's been retweeted.
If it's your tweet you're interested in then you'll get individual notifications in your 'Connect' page each time someone automatically RTs. Eventually (or if you press refresh) the individual notifications are recombined to show the tweet once with all of the RTs and favourites it's received so far. That tweet will be greyed out so you can't expand it yourself, but you can easily count the numbers.
You'll also get notifications if someone manually RTs your tweet (by copying / pasting and perhaps adding a comment) - but you'll have to add that to the auto-RT total.
Here's an example, from the day I started drafting this post, in which I was surprised to find myself retweeted by the Royal Air Force about a job of theirs I'd spotted.
This tweet has been RTed six times but eventually it will scroll down my mentions page and be harder to find. If it receives another RT later it will pop back up again but if you want to be able to check how it's doing at any time make a note of its address - you can find that in the bit that says 15h just before the text of the tweet starts (that's the time stamp of the tweet, and obviously the time since you sent the tweet will vary but the underlying URL won't). Ignore the bit saying 14h on the top right.
The URL for this particular tweet is https://twitter.com/JoBrodie/status/320098630407163904
and if you go to that page you can see it's since been RTed another time making seven in total, and favourited three times (edit: now back to two, so one person's unfavourited it).
My 'most popular' tweet (according to Favstar) was retweeted 72 times and favourited 18 times - this one is a bit of an outlier!
I was pretty pleased to discover Scottish wikipedia today: "Science is a wey tae find oot things" sco.wikipedia.org/wiki/ScienceIf it's someone else's tweet that you want to keep your eyes on it you can get its URL from the timestamp in the same way, and you can visit its page to see how it's doing - here's the URL for my Scottish wikipedia tweet: https://twitter.com/JoBrodie/status/182559448697552896
— Jo Brodie (@JoBrodie) March 21, 2012
2. Search for at-mentions of the tweet sender, to pick up manual RTs
Annoying people like me who prefer to add comments to a tweet they're retweeting will muck up your numbers. Everyone who clicks on the 'RT' button will be automatically added in Twitter's own count, however 'manual RTs' won't be so you'll need to count them.
Search for their @name (you're going to have to do this within a reasonable timeframe of the original tweet being sent, if days or weeks later you probably won't find them) and look for manual retweets, an example below:
Note: It is also possible that Sh4zny's tweet has itself been retweeted (though it actually hasn't) and you'd need to click on her tweet (or visit its page from the URL in the timestamp) to see this.
For completists you can even try searching for the URL of the tweet itself (as opposed to the URL in the tweet, though you can search for that too) and see if someone's posted a 'have a look at this' tweet, often done to avoid alerting the sender of the tweet that their tweet is being commented on. (Unless they know how to look for it of course).
Here's an example of me linking to an older tweet of mine...
This is a really useful tool that lets you see the most popular tweets from someone (in terms of retweets and favourites) and also recent ones. Most of it is free and doesn't require you to log in but there is a Pro version (I've got a trial version of the pro account as a freebie, this is my conflict of interest statement!)
Favstar.fm is the basic address to which you can add /users/screenname to see the tweets of a particular individual (I'm http://favstar.fm/users/JoBrodie/ and Stephen Fry is http://favstar.fm/users/StephenFry) - by default the 'best of' page is shown first.
Because Favstar only captures tweets that have been favourited or retweeted it makes it a lot easier to browse within a much smaller pool of tweets.
Favstar's address formatting is the same as Twitter's so
http://favstar.fm/users/JoBrodie/status/182559448697552896 is the same as
It also tells you what other tweets someone has posted that have been favourited or retweeted.
4. If tweet contains an address (URL) search for that and count any appear
In the example in (2) above I've referred to an earlier tweet of mine and also to a lipid-related blog. There are three URLs going on there - the URL of the tweet as you see it above, the URL of the earlier tweet that I've mentioned in this tweet and the URL of the blog.
Again, this is really for completists but you can search for the basic root of a website (http://xyz.com) and this will bring up any tweets mentioning (http://xyz.com/blah/blah.php etc).
Incidentally, I assume all business have a search saved on Twitter for their http://www.company.com address (which will pick up any tweet mentioning their company website). Sometimes I'm reminded of this when I tweet something about widgets.com and lo and behold a day or two later Widgets UK favourite it.
Sporadically brilliant this one. You can find a fairly random selection of tweets but I have always found it to be pot luck. You can also see where links have been posted on Google Plus.
Hope that helps, remember this is just a handful of ways to find out a bit more about probably a small number of tweets and to do so without spending too much money or time. If you need to get information ('metrics') for how you or your company are doing on Twitter, eg to measure your return on investment as well as reach, then you might need to cough up some cash and use a paid-for service. Alas I know next to nothing about this.