Yes you can upload videos from your desktop. Here's how:
To upload and Tweet a video on twitter.com:
Click the Tweet button.
Click Add media.
Choose a video file stored on your computer and click Open.
Note: You’ll be prompted if the selected video is not in a supported format. Maximum file size is 512MB. See here for more details about formats.
In the Edit window, you can trim the length of your selected video between 1 and 30 seconds.
Note: Maximum video length is 30 seconds.
Click Done when you’re finished with your edits.
Complete your message and click Tweet to share your Tweet and video.