The X, C, and LM refer to the number of parts of the dilutent
(water and or alcohol) that one part of the mother tincture is diluted
in. The letters are the roman numerals so X, C and LM refer to
10, 100, and 50,000 respectively. The numbers refer to the number
of times the tincture is diluted by that amount, and succussed between dilutions.
So for example, 6x = 1 part in 10, 6 times. So the final concentration
is 1 in 10 x 10 x 10 x 10 x 10 x 10, or 1 in 1,000,000.
The letter M breaks with the tradition, and refers to a 1000C dilution, not one part in one thousand.
Perhaps counter-intuitively, homeopathic remdies get more effective as they get more diluted. It makes more sense if you consider that the remedy you are taking is the one that would cause your symptoms if you took it 'neat'.
Occasionally a letter 'H' or 'K' is appended to the potency (for example 30CH, or 200CK). This refers to the production method and in most cases is safely ignored.
One further thing to note is that, on mainland Europe, the D scale breaks with the convention of using Roman numerals, and D denotes Decimal (1/10) potency, rather than the 500 scale. i.e. 6X in English potency would be D6.
That's the theory, for the practice, see our Which potency to take page.