The uniform distribution simply states that the probability is the same everywhere. If you knew that there was a single ball on a pool table, but had no idea where it was placed, you could represent its position with a uniform distribution. The uniform distribution usually represents situations such as this, where no useful information is available.
The normal, or Gaussian, distribution is probably the most important distribution for the sciences. The normal distribution represents random variation around an average value. Accuracy when throwing darts at a dart board, a person's height and the average speed of air molecules in a room are all spread out according to a normal distribution. When scientists report on their research, repeated measurements will usually fall into a normal distribution. When this does happen, the average value of the distribution is taken to be the measured value and the uncertainty is the width of the bell at half the maximum height. This distance across is also called standard deviation. The ubiquity of this distribution is the reason for the term "normal".
The exponential distribution is useful for processes that slow down over time. The most famous example is radioactive half-life. Half-life is the amount of time it takes for half the radioactive atoms in a given substance to decay; after one half-life half remains and after two half-lives, one quarter remains.
The probability density of a particular radioactive atom not decaying and a function that tells the amount of remaining radioactive material as a function of time are logically identical. Both functions are exponentials that decay with time.
Distribution functions do not have to be continuous functions like the ones above. A piecewise function is defined by different functions along different intervals. Piecewise distributions come into play when there is an abrupt change in behavior at a given point. If the pool table example, the probability distribution of finding the pool ball somewhere in the entire room would be piecewise: It would be a uniform distribution on the pool table and zero everywhere else.