If a random variable (RV) has a distribution with finite variance (a uniform qualifies), then sums of those random variables converge in distribution to the Normal/Gaussian distribution as the number of terms in the sum increases. This is known as the central limit theorem. The rate of convergence depends on whether the underlying RVs are continuous or discrete, symmetric or skewed, and independent or correlated. Independent observations from symmetric continuous distributions, such as the uniform, have a rapid rate of convergence.
A sum or average of two independent uniforms has a symmetric triangular distribution. I don't know any name for larger sums other than "convolution of k uniforms", but the larger k is the more it gets curvy in the middle and tapered towards the ends and looks like a normal. By the time you get up to k = 12, you need a fairly large sample size to be able to reject the hypothesis that the sum or average is normal.
If you are averaging k independent Uniform(0,1)'s, the expected value of the average will be 0.5 and the variance will be 1/12k.