What is the "equivalent" of normal distribution in an interval? What is the most "natural" family of distributions on an interval [0,1] indexed by their mean $\mu$ and standard deviation $\sigma$? I am looking for something occurring in "nature", like normal distributions, except that it has to take values from 0 to 1.  
 A: The problem is that the normal has so many properties that might lead someone to consider it natural for this or that problem that we're left to ponder which properties are most critical. 
While I here attempt to answer the question at face value, when choosing a distributional model on the unit interval (or indeed in any other case), I'd strongly urge considering the point in whuber's comment under the question.
There's no really 'natural' candidate that's typically indexed by $\mu$ and $\sigma$, though there are two parameter families on the unit interval which have a mean and variance that are functions of the more usual parameters.
 
The Beta distribution family
A very widely used family of two-parameter continuous distributions on the unit interval is the beta family. It should be possible to reparameterize in terms of $\mu$ and $\sigma$, but it would be considerably less 'nice' looking that way.
$$f(x;\alpha,\beta) = \frac{1}{\text{B}(\alpha,\beta)} x^{\alpha-1}(1-x)^{\beta-1};\quad 0\leq x\leq 1,\alpha,\beta>0$$
It has $\mu = \frac{\alpha}{\alpha+\beta}$ and $\sigma^2 = \frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}\,$.
 
Maximum entropy distribution
The maximum entropy distribution with fixed mean and variance on a closed interval appears (via a theorem of Boltzmann's) to be a truncated normal. 
Truncated normals are sometimes used in various applications, and are indexed by $\mu$ and $\sigma$*, but I wouldn't say they were usually regarded as the most natural for many problems.
 * but beware! In the truncated normal written in the usual way, the parameters $\mu$ and $\sigma$ aren't the mean and standard deviation of the truncated variable, but of its untruncated parent.
Interestingly, while the beta included the uniform as a special case, the truncated normal includes it as a limiting case.
 
Given the phrasing of your question, those would be the most obvious candidates, and of those, the most widely applied is no doubt the beta family. 
