i believe your phone might be infected with virus. Have you tried scanning the bothe internal and external memory with an antivirus? if not, download an antivirus program form google play and then scan the phone to remove the viruses.
Do you know the type of video your phone can play?...
Most phone can play videos in 3gp an mp4 but not flv, mpeg, mpeg2 or 3, wmv, etc if your videos were downloaded from youtube, it might be that they are in flv format. You must convert them to either 3gp or mp4 to see if it will play.
How many video players do you have on your phone? do you know that when you install a video player, you also need to install the video codecs that can help the video player to decode the video file so that it can play it?
You should go to google play and install popular video players like mx player and also type vedeo codecs in the search box. install some for 3gp, mp4 mpeg,wmv etc....as may as you want.
If you do these and you still can play video, let us know