Over the years, many dedicated and talented people have developed statistical software packages and made them freely available worldwide. Although some of these programs may not have the scope of coverage or the polish of the commercial packages, they're high-quality programs that can handle most, if not all, of what you probably need to do.
OpenStat and LazStats
OpenStat, developed by Dr. Bill Miller, is an excellent free program that can perform almost all the statistical analyses you will encounter. It has a very friendly user interface, with menus and dialogs that resemble those of SPSS. Dr. Miller provides several excellent manuals and textbooks that support OpenStat, and users can email him directly to get answers to questions or problems they may have. OpenStat runs on Windows systems.
An alternative is LazStats, also from Dr. Miller, which has many of the same capabilities as OpenStat but can run directly (without emulation) on Macintosh and at least some Linux systems.
Fun fact: The "Laz" in "LazStats" doesn't stand for lazy; it stands for the free Lazarus compiler that was used to create the software for Mac and Linux as well as Windows.
R statistics software
R is a free statistical programming and graphical system that runs on Windows, Macintosh, and Linux systems. It's one of the most powerful computing software packages available, with capabilities surpassing those of many commercial packages. It has built-in support for many kinds of statistical analysis, and many hundreds of add-on packages (also free) extend its capabilities into every area of statistics.
You can generate almost every imaginable kind of graph with complete control over every detail.
But R is not easy to use. Its user interface is very rudimentary, and all analyses have to be specified as commands or statements in R's programming language (which is very similar to the S language used by the commercial S-Plus package). It may take you awhile to become proficient in R, but once you do, you'll have almost unlimited capability to carry out any kind of statistical analysis you can think up.
Epi Info
Epi Info, developed by the Centers for Disease Control, was designed to be a fairly complete system to acquire, manage, analyze, and display the results of epidemiological research, although it's useful in all kinds of biostatistical research.
It contains modules for creating survey forms, collecting data, and performing a wide range of analyses: t tests, ANOVA, nonparametric statistics, cross tabulations, logistic regression (conditional and unconditional), survival analysis, and analysis of complex survey data. Epi Info runs under Windows.
PopTools
PopTools is a free add-in for Excel, written by Greg Hood, an ecologist from Australia. It provides some impressive extensions to Excel — several statistical tests (ANOVA, chi-square, and a few others), a variety of matrix operations, functions to generate random numbers from many different distributions, programs that let you easily perform several kinds of simulations (bootstrapping, Monte-Carlo analysis, and so on), and several handy features for checking the quality of your data.
Definitely worth looking at if you're using Excel on a Windows PC (unfortunately, it doesn't work with Mac Excel).
PS (Power and Sample Size Calculation) and G*Power
The PS program, from W.D. Dupont and W.D. Plummer of Vanderbilt University, does a few things, and it does them very well. It performs power and sample-size calculations for Student t tests, chi-square tests, several kinds of linear regression, and survival analysis.
It has a simple, intuitive user interface and a good help feature, and it provides a verbal description of the analysis (describing the assumptions and interpreting the results) that you can copy and paste into a research proposal or grant application. You can create graphs of power versus sample size or effect size for various scenarios and tweak them until they're of publication quality.
Another excellent power/sample-size program is G*Power. This program handles many more types of statistical analyses than PS, such as multi-factor ANOVAs, ANCOVAs, multiple regression, logistic regression, Poisson regression, and several nonparametric tests. Like PS, it also provides excellent graphics.
G*Power can be more intimidating for the casual user than PS, but because both products are free, you should download and install both of them.