what does the length of each bar represents, in the context of discrete uniform distribution? In probability theory and statistics, the discrete uniform distribution is a symmetric probability distribution whereby a finite number of values are equally likely to be observed; every one of n values has equal probability 1/n.
Page 68 of "Kevin Patrick Murphy. Machine Learning: A Probabilistic Perspective." consider following figure as "uniform distribution". 


For illustration purposes, let us use a simple prior which puts uniform probability on 30 simple arithmetical concepts, such as “even numbers”, “odd numbers”, “prime numbers”, “numbers ending in 9”, etc. To make things more interesting, we make the concepts even and odd more likely apriori. We also include two “unnatural” concepts, namely “powers of 2, plus 37” and “powers of 2, except 32”, but give them low prior weight. See Figure 3.2(a) for a plot of this prior. We will consider a slightly more sophisticated prior later on.

what does the length of each bar represents?
Note: the sum of all of 32 items is equal to 1.0
first item represents 0.131510
3rd item represents 0.000263
what does this mean? 
 A: The graph is just a probability mass function (pmf), so the length of each bar indicates the prior probability that people ascribe to each of the "concepts" listed along the y axis. This particular example is a little confusing out of context. 
That context is Josh Tannenbaum's PhD Thesis "A Bayesian framework for concept learning", and is related[*] to an experiment he calls "the numbers game". Participants play with a computer,which chooses a rule or concept that generates numbers between 1-100 (e.g., "even numbers" generates 2,4,6,...,100). It doesn't tell the participants the rule itself, but instead shows them one of the generated examples, like {16}. Subjects are then asked to guess another number that the computer could also generate.  Someone thinking that the rule is "powers of two" might say {32}; someone else expecting numbers beginning with 1 might respond with {100} or {11} instead. The computer then shows each participant more numbers, like {4,8, 32}, and they are asked to refine their guesses.  
In the numbers game, people favor concepts like "the set of all odd (or even) numbers" over others, which influences their guesses. Having seen just {16}, people tend to guess any numbers between 2-32, but after seeing {16,2,8,64}, guesses on even numbers are vastly more common. Tannenbuam argues that this is driven by a Bayesian-like updating process, which starts from a set of "intuitive" priors and refines them based on the evidence and internal 'simulations' of which could happen. A lot of his work is interesting and well-written, if this sort of thing strikes your fancy. 

[*] To be clear, this particular plot isn't real data; it's a hypothetical version he uses as an example, but it does match what he later finds experimentally.
A: This chart represents several different uniform distributions. Even and Odd numbers are each half of all integers, so I would expect their bars to each represent a length of 0.5. 
The "multiples of X" are their own distribution, the "ends in X" are their own distribution, and so on. 
The length of the bars still represent the probability of that event, there are just multiple distributions thrown onto one chart.
