Wacom pens have a button on them that can be used for both left-click and right-click mouse functions, so yes, you can use the pen for a "click on the screen" if what you really mean is a left/right click function. Also, tapping the pen on the tablet serves the same purpose as a right click.
The Cintiq is very pricey, so to better answer your question, the first thing I'd ask is what is your budget? The Graphire series of tablets are useful but have a lower resolution and fewer pressure levels if I remember correctly, while the Intuos tablets offer higher resolution/more pressure levels but for a higher price. My old Art Pad II is somewhat comparable to the Intuos and I like the 4x5 pad size since it is small enough to not crowd my desktop. But, many users may favor the next larger size. I think the Intuos tablet in a 4x5-inch size (active pad area, not total tablet size) runs around $200, maybe a bit less.
On the whole, nearly anything you perform with a mouse can also be performed with a pen & tablet, but there is some learning involved in getting used to how to use the new tools. Higher-price tablets also have on-tablet hotkeys for function keys and other shortcuts I believe. You can't go wrong in purchasing a tablet except perhaps by spending more for one than your needs really justify.