This is the simplest and most familiar representation of a fraction. A slice of pizza or a section of a chocolate bar are common images used to explain fractions to beginning learners. The basic idea is that of cutting something into an equal number of pieces and expressing one or more of those pieces as a part of the whole. More abstract fractions such as a quarter of an hour also represent this kind of fraction.
This is another concrete way fractions are represented to beginning learners. A fraction as a part of a set involves selecting a few items from a group based on different criteria. The group of items may or may not be identical. If you select five red beads from a group of 20, this represents the fraction 5/20 or 1/4. Alternatively, you may have a group of multi-colored beads and select just the green beads. This representation of a fraction lends itself well to demonstration using manipulatives.
A fraction can also represent a ratio which is essentially a comparison between two different numbers. If I have 1/5 as many books as you do, it means that the ratio of the number of books I have to the number of books you have is 1:5 or that you have five times as many books as I do. The advantage of using a fraction notation for a ratio is that it can be used to perform mathematical operations.
A fraction can simply represent a division. One way to implement this division would be to convert a fraction to a decimal. For example 4/5 represents the division of 4 by 5. Doing this division in terms of decimals we get 4/5 = 0.8. Students often do not understand that a fraction implies division and are confused when they hear phrases such as "1/4 of" which implies division by four.
A fraction, finally, is a point on the number line. A fraction is a rational number that has a certain position on the number line. For instance, the fraction 3/5 can be placed at an exact location on the number line between 0 and 1. This is a more abstract interpretation of a fraction but is useful for computations.