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What datasets exist out on the internet that I can run statistical analysis on?

The datasets package is included with base R. Run this command to see a full list:

library(help="datasets")

Beyond that, there are many packages that can pull data, and many others that contain important data. Of these, you may want to start by looking at the HistData package, which "provides a collection of small data sets that are interesting and important in the history of statistics and data visualization".

For financial data, the quantmod package provides a common interface for pulling time series data from google, yahoo, FRED, and others:

library(quantmod)
getSymbols("YHOO",src="google") # from google finance 
getSymbols("GOOG",src="yahoo") # from yahoo finance 
getSymbols("DEXUSJP",src="FRED") # FX rates from FRED

FRED (the Federal Reserve of St. Louis) is really a landmine of free economic data.

Many R packages come bundled with data that is specific to their goal. So if you're interested in genetics, multilevel models, etc., the relevant packages will frequently have the canonical example for that analysis. Also, the book packages typically ship with the data needed to reproduce all the examples.

Here are some examples of relevant packages:

alr3: includes data to accompany Applied Linear Regression (http://www.stat.umn.edu/alr)
arm: includes some of the data from Gelman's "Data Analysis Using Regression and Multilevel/Hierarchical Models" (the rest of the data and code is on the book's website)
BaM: includes data from "Bayesian Methods: A Social and Behavioral Sciences Approach"
BayesDA: includes data from Gelman's "Bayesian Data Analysis"
cat: includes data for analysis of categorical-variable datasets
cimis: from retrieving data from CIMIS, the California Irrigation Management Information System
cshapes: includes GIS data boundaries and data
ecdat: data sets for econometrics
ElemStatLearn: includes data from "The Elements of Statistical Learning, Data Mining, Inference, and Prediction"
emdbook: data from "Ecological Models and Data"
Fahrmeir: data from the book "Multivariate Statistical Modelling Based on Generalized Linear Models"
fEcoFin: "Economic and Financial Data Sets" for Rmetrics
fds: functional data sets
fma: data sets from "Forecasting: methods and applications"
gamair: data for "Generalized Additive Models: An Introduction with R"
geomapdata: data for topographic and Geologic Mapping
nutshell: contains all the data from the "R in a Nutshell" book
nytR: provides access to congressional vote data through the NY Times API
openintro: data from the book
primer: includes data for "A Primer of Ecology with R"
qtlbook: includes data for the R/qtl book
RGraphics: includes data from the "R Graphics" book
Read.isi: access to old World Fertility Survey data

A broad selection on the Web. For instance, here's a massive directory of sports databases (all providing the data free of charge, at least that's my experience). In that directory is databaseBaseball.com, which contains among other things, complete datasets for every player who has ever played professional baseball since about 1915.

StatLib is an other excellent resource--beautifully convenient. This single web page lists 4-5 line summaries of over a hundred databases, all of which are available in flat-file form just by clicking the 'Table' link at the beginning of each data set summary.

The base distribution of R comes pre-packaged with a large and varied collection of datasts (122 in R 2.10). To get a list of them (as well as a one-line description):

data(package="datasets")

Likewise, most packages come with several data sets (sometimes a lot more). You can see those the same way:

data(package="latticeExtra")
data(package="vcd")

These data sets are the ones mentioned in the package manuals and vignettes for a given package, and used to illustrate the package features.

A few R packages with a lot of datasets (which again are easy to scan so you can choose what's interesting to you): AER, DAAG, and vcd.

Another thing i find so impressive about R is its I/O. Suppose you want to get some very specific financial data via the yahoo finance API. Let's say closing open and closing price of S&P 500 for every month from 2001 to 2009, just do this:

tick_data = read.csv(paste("http://ichart.finance.yahoo.com/table.csv?",
    "s=%5EGSPC&a=03&b=1&c=2001&d=03&e=1&f=2009&g=m&ignore=.csv"))

In this one line of code, R has fetched the tick data, shaped it to a dataframe and bound it to 'tick_data' all . (Here's a handy cheat sheet w/ the Yahoo Finance API symbols used to build the URLs as above)

http://www.data.gov.uk/data

Recently setup by Tim Berners-Lee

Obviously UK based data, but that shouldn't matter. Covers everything from abandoned cars to school absenteeism to agricultural price indexes

Have you considered Stack Overflow Data Dumps?

You are already familiar with what the data represents i.e. the business logic it tracks

A good start to look for economic data are always the following three addresses:

World Bank - Research Datasets
IMF - Data and Statistics
National Bureau of Economic Research

A nice summary of dataset links for development economists can be found at:

Devecondata

Edit:

The World Bank decided last week to open up a lot of its previously non-free datasets and published them online on its revised homepage. The new internet appearance looks pretty nice as well.

The World Bank - Open Data

Another good site is UN Data.

The United Nations Statistics Division (UNSD) of the Department of Economic and Social Affairs (DESA) launched a new internet based data service for the global user community. It brings UN statistical databases within easy reach of users through a single entry point (http://data.un.org/). Users can now search and download a variety of statistical resources of the UN system.

http://www.data.gov/ probably has something you can use.

In their catalog of raw data you can set your criteria for the data and find what you're looking for http://www.data.gov/catalog/raw

A bundle of 268 small text files (the worked examples of "The R Book") can be found in The R Book's companion website.

You could look on this post on FlowingData

Collection of over 800 datasets in ARFF format understood by Weka and other data analysis packages, gathered in TunedIT.org Repository.

See the data competition set up by Hadley Wickham for the Data Expo of the ASA Statistical Computing and Statistical Graphics section. The competition is over, the data is still there.

UC Irvine Machine Learning Repository has currently 190 data sets.

The UCI Machine Learning Repository is a collection of databases, domain theories, and data generators that are used by the machine learning community for the empirical analysis of machine learning algorithms.

I've seen on your other questions that you are apparently interested in data visualization. Have then a look at many eyes project (form IBM) and the sample data sets.

Similar to data.gov, but european centered is eurostat

http://epp.eurostat.ec.europa.eu/portal/page/portal/statistics/search_database

and there is a chinese statistics departement, too, as mentioned by Wildebeests

http://www.stats.gov.cn/english/statisticaldata/monthlydata/index.htm

Then there are some "social data services" which offer the download of datasets, such as swivel, manyeyes, timetric, ckan, infochimps..

The FAO offers the aquastat database with data with various water related indicators differentiated by country.

The Naval Oceanography Portal offers, for instance, Fraction of the Moon Illuminated.

The blog "curving normality" has a list of interesting data sources.

Another collection of datasets.

Here's an R package with several agricultural datasets from books and papers. Example analyses included: agridat