A common grammatical feature that all the Celtic languages share. This write-up will focus almost exclusively on Irish, being that that's the only Celtic language with which I am personally familiar.
The process involved is that, simply, words in the language will begin with a certain sound or phoneme, and that in certain situations, this sound will mutate or change slightly (or not so slightly) into a different but similar sound. Such changes are common in the phonology of most languages; for example, in English, the word house ends in an "s" sound, but in the plural form houses, this sound changes to a "z" sound. The distinction is that this is what is known as a phonologically-conditioned sound change; the "s" sound in houses is in a phonological environment which changes it to a "z" sound. (The "s" phoneme is placed in between two vowels, and vowels are always voiced- articulated in such a way that the vocal cords are vibrating- so the "s" sound itself becomes voiced, ie. it changes to a "z".)
The phonological changes in the Celtic languages, though, are gramatically-conditioned, which is a far more rare and unusual process. That's to say that it's not the phonological environment which changes the sounds, but the presence or absence of certain grammatical factors, eg. whether a verb is in the present or past tense, or whether the noun follows a certain combination of prepositions. Here are some examples from Irish:
The word bris (pronounced brish) is the verb root meaning "to break". It is of course conjugated to reflect the person, number, etc. For example, brisim (brishem) "I break", briseann sé (brishen shay) "he breaks", etc. But in the past tense, the word-initial mutation applies, and the initial "b" sound changes to a "v" sound. (Both "b" and "v" are labial sounds- sounds made with the lips- but "b" is an occlusive or a "stop", a sound in which the airflow is completely stopped briefly, while "v" is a fricative or continuant, a sound in which the airflow continues throughout the articulation.) So, bhris mé (vrish may), "I broke". This type of sound change is known in Irish as "aspiration", and it usually involves a stop changing into a continuant with a similar place of articulation.
The second type of sound change in Irish is known as "eclipsis". Here's an example with the word "bord" (pronounced board), meaning "table".
bord = a table
ar an (air on) = on the
but when joined, eclipsis applies, and they become:
ar an mbord (air on moard), "on the table".
Whereas aspiration usually involves a stop becoming a continuant, or a "hard" sound becoming a "softer" sound in very loose terms, eclipsis often involves a stop becoming nasalized. (In the example above, both "b" and "m", again, are labial sounds, but "b" is a pure stop, while "m" is a nasal sound as some airflow continues through the nasal passage even though the mouth is fully closed.)