What are you taking photos of?
The AF won't focus if whatever you're looking at is too close to the camera and if the AF won't focus then the camera won't let you take the photo. You may notice in the viewfinder (somewhere) a green circle that lights up when it's focussed. If it keeps flashing then the AF hasn't found a focus point.
You may also find that the AF won't work if you're looking at something that's not very contrasty, or there's not enough light.
In AF does the camera try to focus? Can you hear it, or see the focus ring moving back and forward?
Depending on the lens, some don't like it if you try and focus manually when you're in AF and you can break the focus mechanism.
You can takes photos in manual focus because the camera is relying on you to focus properly.
Again, what are you taking photos of & what shutter speed are you using? With MF, if what you're looking at is too close you won't be able to focus, lenses have a minimum focus distance. Move back until things are in focus.
If you're handholding with a slow shutter speed you'll get camera shake/motion blur.
When you are focussing manually is the image sharp in the viewfinder?