First, let`s look at a game of tenpin bowling:

The numbers in the top row are the frame number, the second row is the pins knocked down (pinfall) and the third row is the actual score.  Now, let`s look at what all of the "x"`s and "/"`s mean:

1. A Strike (X) is worth 10 pins plus the number of pins knocked down on the next two balls.
2. A Spare (/) is worth 10 pins plus the number of pins knocked down on the next ball (one ball only - not frame)
3. An Open Frame (one where not all pins are knocked down after both balls have been bowled) has no bonus score.
4. If a Strike is bowled on the first ball of the tenth frame, the bowler receives two extra balls.
5. If a Spare is bowled on the first two balls of the tenth frame, the bowler receives one extra ball.
6. There is a maximum of three balls in the tenth frame so the score in the tenth frame is simple totaled and added to the score from the ninth.
7. Note that the circle around the 7 in the sixth frame indicates a split.  A split can also be denoted by the letter "s"as in the examples below.

Note that scoresheets can come in a variety of display formats, but they all mean the same thing. 

Scoresheet - one of the standard formats with triangles and squares to indicate spares and strikes.  Note the circle around the 7 in the sixth frame denoting a Split.

Scoresheet - exactly the same score as above but with "/"`s and "X"`s instead of triangles and squares.  Note the "s"next to the 7 in the 6th frame indicating a split.

Scoresheet - again same as above, but just showing "marks"(Strikes, Spares, Miss only - no pinfall).  This is easy to read, but doesn`t show pinfall, so a miss could be a one-miss or a nine-miss.

Now looking at the game above, we can add up the scores frame by frame:

1. In the first frame, the score is 10 (because we got a spare) plus the next ball (8), totaling 18.
2. Second Frame: Score from first frame (18) plus eight plus 1.  Equals 27.
3. Third Frame. Score from second frame (27) plus ten (because we got a spare) plus the next ball (4).  Equals 41.
4. Fourth Frame. Score from 3rd (41) plus four plus five.  Equals 50.
5. Fifth Frame. Score from 4th (50) plus ten (spare) plus next ball (X = 10).  Equals 70.
6. Sixth Frame. Score from 5th (70) plus ten (X) plus next two balls (X and X = 10 + 10).  Equals 100.
7. Seventh Frame. Score from 6th (100) plus ten (X) plus next two balls (X and 7 = 10 + 7). Equals 127.
8. Eighth Frame. Score from 7th (127) plus ten (X) plus next two balls (7 and - = 7 and 0).  Equals 144.
9. Ninth Frame. Score from 8th (144) plus seven plus 0 (- = miss = 0). Equals 151.
10. Tenth Frame. Score from 9th (151) plus ten (9/) plus ten (X).  Equals 171.

Remember that in the tenth frame, simply total the frame and add it to the ninth to get the final score.  There is no extra bonuses. Sometimes you will see a circle around a number in the score.  This means a SPLIT has occurred.  A split is when the head pin has been knocked down and the pins remaining are separated by at least one pin.  This sounds rather confusing, but for example, it is possible to get a "nine split".  This means that the head pin (pin 1 - the closest to the bowler) was knocked over exposing the 2 and 3 pins (the next row back after the head pin).  Since the 2 and 3 pins are separated by more than one pin (the head pin is no longer there), this is considered to be a split. The most common splits are 4-6, 4-6-7-10, 7-10, 4-7-10, 6-7-10, 3-9, 2-7 but there are literally hundreds of possible splits.  One important thing to note is that it is not possible to have a split if the head pin is present - if the head pin is present then no matter what other pins have been knocked down, it is not called a split. Just because it is not called a split doesn`t mean it isn`t difficult to get some spares when the head pin is left with other pins.  For example, the 1-7-10 combination is not called a split but is also not an easy combination to spare. You may also see some other marks, such as W, meaning washout.  A Washout is basically the same as a split, but the headpin is present.  Not many scoring systems will show you a W, but some do.