How could you miss one of the capital's most iconic areas? Come and marvel at Nelson's Column and the four huge lion statues. Feeding the pigeons is now discouraged (due to the spread of diseases), so please don't bring them any treats.
On the north side of the square you can visit the National Gallery and just around the corner on St. Martin's Lane is the National Portrait Gallery. Both have free permanent displays and regular special exhibitions.
Trafalgar Square was designed by John Nash in the 1820s and constructed in the 1830s. It is both a tourist attraction and the main focus for political demonstrations. There are some great extras to spot: see this video of hidden gems in Trafalgar Square. Look out for the George Washington Statue and the World's Smallest Police Box, as well as the London Nose.
Within walking distance of Trafalgar Square you can easily go shopping in Covent Garden, have a meal in Chinatown, walk down Whitehall to Parliament Square and see the Houses of Parliament and Big Ben, or walk down the Mall to Buckingham Palace.