We consider a prior for nonparametric Bayesian estimation which uses finite random series with a random number of terms. The prior is constructed through distributions on the number of basis functions and the associated coefficients. We derive a general result on adaptive posterior convergence rates for all smoothness levels of the function in the true model by constructing an appropriate "sieve" and applying the general theory of posterior convergence rates. We apply this general result on several statistical problems such as signal processing, density estimation, various nonparametric regressions, classification, spectral density estimation, functional regression etc. The prior can be viewed as an alternative to the commonly used Gaussian process prior, but properties of the posterior distribution can be analyzed by relatively simpler techniques and in many cases allows a simpler approach to computation without using Markov chain Monte-Carlo (MCMC) methods. A simulation study is conducted to show that the accuracy of the Bayesian estimators based on the random series prior and the Gaussian process prior are comparable. We apply the method on two interesting data sets on functional regression.