Toilet flushes just fine. But the bathroom sink is backed up
The U bend or "P" trap below the sink may be plugged with hair and soap scum causing it to back up. when you flush the toilet the gurgling could be caused from the slight vacuum the water running down the pipe when you flush. The drain stopper in the sink is usually a place hair and soap get stuck also. 

Place a bucket under the pipe and dissemble the trap. It's usually not very difficult. The water will run into the bucket. Clean out the U and, if you can see hair stuck in the stopper, the shaft that pushes it up and down can be unscrewed and removed. Then the hair usually falls out. Reassemble and see if everything drains. 

The plug could be a few feet further down if this did not work but a inexpensive hand snake of 15-25' put down the pipe after the u bend of the sink would clear the plug if the "P" trap was not the problem. 

This repair is something every home owner should try and it will save hundreds over the years instead of calling a plummer. Here is a link to a nice hand snake There are cheaper and more expensive ones available. If taken care of they do last decades. Good luck on your first plumbing job.