Twitter is very slowly rolling out the option to download and archive of all the Tweets you've made since signing up.
Okay so first of all it appears that newer accounts on Twitter are getting their archives available sooner than oldies. At least for me my much newer and less used accounts (active since 2009 and 2011) have the archives already, whereas my main account (active since summer of 2008) hasn't got it yet.
Here's what you need to do to get your archive:
You must be logged in to Twitter. Go to your account Settings and scroll down the page.
If your archive has been made available there's a new section for it down there. It's titled Your Twitter archive and there's a big button there that says Request your archive. Click this big button.
You'll get a notification (right there on your browser): "We've received your request! A link will be emailed to you when your archive is ready. This may take a little while, so please be patient."
It may take long or short time (as notified), but after a while you'll get an email titled something like: "Your Twitter download is ready, yourTwitterhandle!" You can request the email to be resent in case it doesn't show up or something. Remember that sometimes legit emails end up to your spam box.
Within that email there's a big blue button saying Go now. Click the button.
This opens a web page, which is titled "Your Twitter archive". Click the blue Download-button on that page.
The archive is downloaded as a zip-file, which you will need to unzip. Unzipped it's a folder containing lots of stuff, including a file called index.html. Ignore the other stuff and open that index-file in your browser (most likely it will open to your default browser if you double click it). This is your archive.
The archive includes all of your original Tweets (including @replies etc.) and ReTweets. Tweets are divided in years and months and you can search them.