The width of the cue tip is sized to fit the width of the ferrule. The ferrule is the next bit of material, fitted onto the thin end of the stick. The ferrule width (and by extension, the shaft) can be from 9mm to 14mm in diameter. Snooker cues generally use the smaller tips (9-11mm) and most regular pool sticks use tips that can range from 12-15mm. Thrown into this variety of widths are player preferences - variations of 12.25mm, 12.50mm, 12.75mm, etc..
Cue tip shapes are dome-shaped, This ensures that as much material contacts the cue tip as possible whether the tip touches the cue ball up, down, and left/right sides of the center. This means that the curve of the tip must match the curve of the cue ball. For snooker's smaller balls, this curve is best described as matching the U.S. 10 cent piece (dime). For pool, the recommended curve fits the U.S. 5 cent piece (nickel). This is the silhouette curve of the coin and the tip - referred to as the nickel-curve and dime-curve. (For billiards, with the larger ball, you can use the quarter-curve (US. 25 cent piece).
The smaller, thinner shafts with the correspondingly smaller width (9mm to 10.5 mm) will have the tip with the dime shape. You will see such thin shafts most often used in Snooker. A few players will have thin shafts on their pool sticks because of personal preference. The majority of pool sticks use the larger diameters with the tip curved to the nickel width.
Because there are so many custom widths for shafts, tip manufacturers make all of their sizes in millimeter increments. You can get a 13 mm or a 14 mm, but not a 13.75 mm. To make this work, a 14 mm tip is mounted on the shaft, and then shaved or abraded away until the tip matches with width of the ferrule.
The most common tips are a single leather material. There are some tips made out of layered material. Recently, plastic tips have been introduced, used primarily on break cues. Tips also come in hardness variations - from soft material, medium-soft, medium, medium-hard, hard-medium, hard-hard, etc.
Soft tips tend to mushroom - spread out over the edges of the ferrule. This requires continuous trimming until the material has been compacted enough to be stable. Hard tips tend to slip off the cue ball unless careful attention is paid to chalking. Because of these two limitations, most players tend to use medium tips.
There are a lot of tip choices and manufactures. The decision to experience with a new type of is not a simple choice. It requires the removal of an old tip and adding the new tip, then shaping the curve. Once done, a lot of ball hitting has to be done, comparing the results of the new tip against the experiences of the old tip. Some players don't like the change and go back to the old brand. Others like the change in ball action and become heralds for the brand.
It is a good idea to try several tips. Some tips can fit your style of shooting better than others. The only way to find out is to experiment. This may take some courage on your part - but the lessons learned from trying other tips is a once in a lifetime process. A good tip mechanic can swap out several tips for you on request - at a reasonable price.