That's a bit of an open question without enough info. If I had a similar problem I'd run a disk cleanup, antivirus and clear the cache on my browser. Do other websites work without issue? What exactly happens when you try to watch YouTube? What browser, what operating system?
If the page opens but the video won't display a quick fix from a laptop/PC is to click on the video, wait for the settings to appear then click on the start arrow. You will have to do this for each video.
Youtube have a habit of changing their settings from time to time so you have to find a way round the changes.
What is the laptop running?
If its old, like vista or even older, then it can not cope with utubes latest video format and you will only be able to view much older videos.
try any more modern device and see if you can view with it