The white keys on the piano are the "natural" notes. The black keys are the "sharps" and "flats".
Notice that there are no black keys between E and F and between B and C.

The smallest interval between notes is called a "semitone" (ST).
Where there is a black key between two white keys, the interval between the two white keys is a "tones" (T) (same as 2 semitones).
Where there is not a black key between two white keys, the intervals between the two white keys is a semitone.

If we start on the note C and go up each white key (C D E F G A B C) then we have the scale of C major. The order of tones and semitones is T T ST T T T ST. Most music you have played will have been written in a major key (with this order of tones and semitones) and this is known as the Ionian Mode or 1st Mode. Examples include Hot Cross Buns and  Twinkle, Twinkle, Little Star. Music written in the Ionian Mode sounds happy as the primary chords (I, IV and V) are all major chords.

If instead we start on the 2nd note of the C major scale and play D E F G A B C D then the order of tones and semitones has changed to T ST T T T ST T. This is known as the Dorian Mode or 2nd Mode. Many English folk songs were written using this mode, such as What Shall We Do With The Drunken Sailor, Greensleeves and Scarborough Fair. It's also a popular mode used in jazz. Music written in the Dorian Mode has both major and minor primary chords so sounds both happy and sad. 

If we start on the 3rd note of the C major scale and play E F G A B C D E then, once again, the order of tones and semitones has changed, this time to ST T T T ST T T. This mode is called the Phrygian Mode or 3rd Mode and a well-known example was composed by Thomas Tallis (1505-1585) and 400 years later used by Vaughan-Williams in his Fantasy on a Theme by Thomas Tallis for string orchestra. This mode has a minor feel to it.

If we start on the 4th note of the C major scale and play F G A B C D E F then the order of tones and semitones has again shifted to give T T T ST T T ST. This sounds like an F major scale when you've forgotten to play B flat! This mode is called the Lydian Mode or 4th Mode and an example is the theme from The Simpsons. This mode has a major feel to it.

Starting on the 5th note of the C major scales gives us G A B C D E F G and another order of tones and semitones: T T ST T T ST T. This sounds like a G major scale when you've forgotten to play F sharp! This mode is called the Mixolydian Mode or 5th Mode and is quite common in pop music. An example includes the chorus from Hey Jude by The Beatles where the chord sequence is I VII IV I (G F C G) repeated over and over again. This mode also has a major feel to it.

If we start on the 6th note of the C major scale then we get A B C D E F G A, which is the same as the Natural Minor Scale. The order of tones and semitones is now T ST T T ST T T and this mode is known as the Aeolian Mode or 6th Mode. A well-known example is the Christmas carol God Rest Ye Merry Gentlemen. Unlike music written in a minor key, all three primary chords in this mode are minor chords so the music sounds sad.

With all the modes so far, the 5th note of the scale has always been a Perfect 5th above the 1st note (root) and Chord I has either been a major or minor chord. If, however, we start on the 7th note of the C major scale and play the notes B C D E F G A B then the interval between the 1st and 5th notes is now a diminished 5th and Chord I is a diminished chord. The interval of a diminished 5th is known in the musical world as "the devil's interval" (because it is difficult to sing). This mode is called the Locrian Mode or 7th Mode and an example is the song Dust To Dust, which is all about a grave digger!