Dotted notes are notes with a small dot written after them. A dot increases the length of a note by one half its value. A dotted quarter note is equal to a quarter note tied to an eighth note (quaver). A dotted half note is equal to a half note tied to a quarter note (crotchet). A dotted whole note is equal to a whole note (semibreve) tied to a half note (minim).
In other words if a basic note lasts one beat, the corresponding dotted one lasts one and a half beats. If it last two beats the dotted note lasts 3 beats. Hold the note for its duration plus half of its original length.
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Dotted Quarter Note
A quarter note equals one beat. Half of that length is one eighth note beat. One quarter note plus one eighth note equals one and a half beats. So a dotted quarter note lasts for one and a half beats.
Dotted Half Note
In common time a half note equals two beats. Half of that length equals one beat. Two beats plus one beat equals three beats. So a dotted half note lasts for three beats.
Here’s a large image showing eighth, quarter, half and whole dotted notes.
Double Dotted
As we’ve seen, a single dot increases a note’s value by 50%. But what if there are two dots after the note? In that case the duration of the note is increased by a total of 75% (which is 50% for the first dot and 25% for the second dot). For instance, a double dotted half note has the value of 2 beats plus one beat, plus half a beat, giving you a total of 3 1/2 beats.
For a double dotted quarter note, the duration is one quarter note plus one eighth note plus a sixteenth note, or 1 3/4 beats.
In terms of notation, if the dotted note is on a space, its rhythm dot also goes on the space, but if its on a line, the dot goes on the space above. Same thing applies to notes on ledger lines.
Like notes, rests can also be dotted, increasing their value in the same way. A dotted quarter rest lasts for 1 1/2 beats, a dotted half rest lasts for three beats, and a dotted whole rest lasts for 6 beats. 
Related Reading: Types Of Notes