Do I Need to Play Black Notes?

Simple Answer

You can play thousands of songs without needing to play any black notes. Most songs only use the standard scale notes, so many songs can be played just using white notes, if they are played in the 'key of C'. Occasionally, some songs use 'non-standard' notes which are outside the standard scale notes. When playing in the key of C, this means they will use one or more black notes. Some playing styles such as Blues and Jazz use notes outside the standard scale (in the key of C these are black notes), which give the music a distinctive sound.

The more detailed answer

Most songs only use notes taken from a 'standard scale' which consists of 7 notes, which have a specific sequence of spaces between the notes. A song will be recognisable no matter where you start the scale on the keyboard. Each different starting position is playing in a 'different key'. The scale of 7 notes includes a mix of white and black notes, to accomplish the specific sequence of spaces between the notes, except for the 'key of C' which uses all white notes. This is why the easiest way to play songs is to play them 'in the key of C', which means you don't have to learn the rules about which black notes to use, which is required when you play in any key other than C. 

Basically, the song sheet will tell you if you need to play black notes, as they will show a '+' symbol beside the melody note or will show an alteration to the standard chord number eg 'm' or 'maj' written beside the chord.