The easiest way to extract a clip from a DVD is using DVD Ripper. The program allows you to extract movie clips and export them to the most common file formats. It also gives you the option to resize video and controlling the output quality. The good thing about DVD ripper is that it selects the codecs to use automatically and shields you from excessive detail and confusing options, while producing good results.

1. Download DVD Ripper and install it – Even though this isn't free software AoA media give you a 30 day trial option which will allow you to extract a 10 minute movie – the maximum size for youtube, perfect!
2. Load a DVD by pressing the "+ Load a DVD button" and select a DVD drive.
3. Select Movie Clip and check the box near the title you want to extract from. (See how to know which title to select below).
4. Set the duration of the clip from the dialog box that opens after clicking the clip.
5. Select the file format to convert to (I use avi for standard extractions and flv for youtube)
6. Change any settings (optional) and click start.

How to know which title to select
To know which title to select play the DVD in you favourite DVD player (Power DVD, VLC, Mplayer, Windows Media Player). From the program’s navigation menu locate the clip you want to extract by browsing through the titles. Take note of the title number and select the same title number in DVD Ripper.

Related:
   How to backup a DVD (even if it has copy protection)

References:
   DVD Ripper Help