Academic journals can be difficult for the general public to access because many require paid subscriptions to read the full text of their articles. The most common way of gaining access to such articles is through an institutional subscription, whereby all members of a given organisation gain access to a given set of journals by logging into to their user account. Universities and public libraries both will generally have subscriptions to a wide range of journals, and so visiting either of these institutions will often enable one to download academic articles. Some universities provide an option for members of the general public (i.e. not students) to purchase a yearly subscription permitting them access to a wide range of journals from their home computer. There are also a growing number of open-access journals, which provide free public access to the full text of all their articles without requiring any subscription.
Increasingly, many researchers publish their articles, or preliminary versions of their articles before final publication, on their own personal websites. As such, one can sometimes gain access to these by a simple google search of the article title, particularly looking for .pdf results (as many papers are posted online in pdf form). Google scholar is a very useful search engine for finding results in academic sources, and often will provide links to publicly accessible pdf downloads of the full texts of articles.
The five major English-language academic publishing companies are Springer, Sage, Taylor & Francis, Wiley-Blackwell, and Reed Elsevier. Their websites contain links to the many hundreds journals published by each company, as well as thousands of academic books in a wide range of subject areas. Some of these materials are available for free, though many require a library or personal subscription.
Wikipedia lists of academic journals: useful for finding sources
Google scholar: excellent tool for searching for academic literature and finding freely-available full text pdfs
PLOS journals: PLOS publishes a number of open-access journals covering various topics in biology