Currently PSP owners have not been able to directly use the PSP's web browser to directly link to Youtube and watch any of the videos available on the site. There are currently ways to watch youtube video on your PSP without hacking the firmware.
Using the web browser on your PSP as well as a wifi connection you can visit a site that will allow you to search for and download youtube videos. In your address bar type in http://www.pspube.com.
The page is a standard black background with orange text. You move the arrow down the page until you get to the option of Click Here to Start Searching. Place the arrow on it and press the x button. This new page will display two search bars one listed as “hot topics” and the one below as “advanced search”. If you are looking for a particular video use the advanced search tab. The next page is a terms of service page you need to scroll down and highlight the Continue to Search Results phrase and press the x button. Now you have a page that has returned searches from youtube. On the left hand side are pictures of the video followed by the name of the video. Choose the video you are interested in and move the cursor to the picture and press the x button. Now you have a choice of downloading the video using tubidy or for a hi-resolution version you can download the video using filehd. Follow the instructions and make sure that all videos saved to your video folder on your PSP ends with .MP4. That is the format that PSP video's play in. Some videos will work others will not and thats only because of the resolution of the video the PSP can handle. This method requires nothing but your PSP and a wifi internet connection as well as enough memory.
You can choose to go to the site directly but instead of the regular youtube address you can visit the mobile youtube link. In your PSP's address bar type in m.youtube.com. You can search for videos you want to view. In the mobile youtube link you will place your arrow on the actual name of the video and press the x button. Once the page loads go to your address bar and hit the x button again. The reason will become clear in a moment. Once the page has refreshed you have to enter www.keepvid.com address. Once there you move the arrow into the url box on the home page. The PSP text entry screen will emerge then press the 'SELECT' button two times this will bring up the 'HISTORY' box in the upper left side of the text entry screen. Highlight the history box and press x. This will display your current history. In the history box choose the m.youtube url with the video you want. Press the 'START' button twice. From here you can download the video to your PSP's Video folder. Remember to have your Java settings on when choosing this option.
If you have a computer, mac or PC will do just find you can go to youtube and search for the videos you want to watch on your PSP. Copy the address of the video and use keepvid.com. Enter the youtube video address in the keepvid.com site. Download the MP4 file and save it. Some videos may or may not work. This is where your computer comes in very handy. You can download a free program to convert all files to fit on your PSP. The most popular and easiest to use is PSP Video 9 which works for both MAC and Windows computers. This allows you to make any video playable on your PSP. For Windows only pc's you can use Sony's proprietary software called Media Go. Transfer the converted files into your Video folder on your PSP.
Those currently are the best options to view Youtube videos on your PSP. Try each option and choose the one that best fits your needs. Obviously if you know you will not have a wifi connection available downloading and converting the video is the best option.