The UN is a massive organization composed of smaller agencies, organizations, and projects that all produce data. In many cases the data they provide is hosted on individual websites, which makes it difficult to find and sift through stats.
The UNdata page (http://data.un.org/) is a central repository for statistical data about countries that captures many of these datasets in one place. You can search for datasets or a particular variable or drill down through categories that are arranged by agency and topic. Datasets include population, education, labor, crime, health, technology, trade, and environmental issues. Once you select a particular dataset you can filter by country and build comparative country tables that you can download.
In addition to comparative tables, which show you one variable for many countries, you can also access country profiles which show you many of the variables for one country. Getting to the profiles (which are collectively called the World Statistics Pocketbook) is a little tricky as they are buried on the site – here’s the link – http://data.un.org/CountryProfile.aspx