How to download videos from sites like youtube and vimeo?

to download from youtube, I don't know If it works on Vimeo.

http://file2hd.com/

Free app, asks for donation, I sent in 10.00 but did not have to.
Downloads Youtube vids in mere minutes, even the 2 hour long ones, so you can watch them at your convenience.
The free version downloads MOST UTubes, from a drop down menu which offers "medium"
You have to pay for the additional app that lets you download in any other format ( mp4, etc)
I get by with just medium.

Does not work with vimeo, I too am looking for an app.

that will let you download almost any video from the web.

RealPlayer Video Downloader Is A Safe and Easy Way to Download Video

See a video you like online? Download it with just one click and save it to your computer. It couldn’t be easier. Once you’ve downloaded free videos you can organize them in the RealPlayer library and create playlists. You can also transfer videos to other devices so you can take them with you and watch them wherever you want.

http://www.real.com/realplayer/download-video

It has a video downloader built in. All you have to do is hover your mouse over the rh corner and the save option will pop up. Click on the item with ??? in the file size area and the download manager will take over. You can use it alongside other browsers without altering either one. I have been using it for many years without any problems at all.
http://dl.maxthon.com/mx3/mx3.5.2.1000.exe

http://downloadvimeo.com/

Past the link to the video and you can download the video as an mp4.

Many of the solutions offered require an application to be installed.

I use It will give you a variety of formats to choose from for downloading from lower quality .flv to higher quality .mp4. It also has a further link that will allow you to download a .mp3 of the video, which is great if it is a speech or an album.

what is "It" ?

I went to add the link to the site, and it apparently didn't copy & paste properly on the forum.

Here is the 'It' in question.

http://www.keepvid.com/

Keepvid can be downloaded and used as an application, but it also works very well from any Java equipped browser which is my preference. It supports YouTube, DailyMotion, Google Video, Megavideo, Metacafe and Vimeo downloads. It will also offer a link to a companion site which will make mp3's of any video from those same sites.

https://addons.mozilla.org/en-US/firefox/addon/video-downloadhelper/

For Firefox: http://userscripts.org/scripts/show/13333
For Chrome: http://userscripts.org/scripts/show/57201
For Opera: http://userscripts.org/scripts/show/36453

http://www.bestvideodownloader.com/

Be sure to decline the extras offered during installation or you'll end with a lot of unnecessary crap.

Offliberty.com is fantastic. Works with youtube, soundcloud, and all other kinds of sites with streaming content. All you do is plug in the page's address and wait for the download link.