In harmony, an applied chord is a chord, not in the key of the melody, that functions as a dominant, subdominant, etc. to another applied chord or a chord in the key of the melody. For example:
Consider the progression C F C D G C
In this example (which is one of the most common forms of applied dominance), the D+, though not within the key of C (i.e. it should be D-), acts as the dominant of G, the dominant of the key of C. In harmony this is progression would be notated: I IV I VV V I (where VV literally means the dominant of the dominant). An example of this kind of applied dominant is the line "Glorious and Free" in O Canada. Another applied dominant chord (although not scale degree II as above) can be heard in the first line of the verse of Maxwell's Silver Hammer.
Another common applied chord is the use of I7 to go to IV. Consider the progression: I I7 IV V I. A great example is the song Something by The Beatles which goes: I Imaj7 I7 (or V7IV) IV V7V VI
For those more familiar with harmony, we can deconstruct this progression as the tonic (C) descending chromatically (Cmaj7) to the dominant of the subdominant (C7), then the dominant of the dominant (D7) resolving to the dominant (G) which forms a deceptive cadence by resolving to the submediant (Am).
Applied chords can also be stacked so that one applied chord is actually the dominant of another applied chord. This sounds confusing but we hear it quite often. A great example of this sort of progression is the song Ophelia by The Band which goes:
C E7 A7 D7 F G or harmonically: I-V7 of V of V of V-V7 of V of V-V7 of V-(IV)-V. Here we consider the F to be a passing chord between V7V and V, hence the brackets.
Since applied chords will always contain notes not in the key (and thus require accidentals), special care must be given when approaching them. Since melodic augmented seconds are considered dissonant in harmony and melodic tritones are also frowned upon in counterpoint, arrangers must be careful to voice these chords properly. There are no real rules as to the chords that precede applied chords hence many dissonant melodic intervals might be heard.
Note also that applied chords function in major and minor keys and can be major, minor, augmented, and diminished although the vast majority of used applied chords are major chords, seventh chords, and diminished or diminished seventh chords since these chords can contain a tritone.
The most common applied chords are I7, VV, viiV, Vvii, and viivii which in C major are: C7, D, F#dim, F#, and A#dim all of which may just as often be used as seventh chords.