Before Twitter introduced its archiving system, there was no way to download all your personal information from their servers and save it on your computer.
There was a workaround, though; but you had to live in the European Union to request it.
If it was your case, you had to follow these instructions, and wait a few days for Twitter to send you your tweets in an archive.
Then, only had to clone this project on your server or computer, and follow the instructions on the Github repo to get a nice SQLite database containing your tweets!
This script will not work with the newer archives provided by Twitter via their website.