Lenition (lat. lenere = weaken) means that a consonant is spoken without a stop of the flow of air i.e. with breath or aspirated. It is because of this that lenition is sometimes referred to as aspiration.

Aspiration in its own right is actually a much more subtle alteration of a phoneme (a p is still a p, even when aspirated). A classical Greek ph was aspirated, but remained a p was no f (it was first much later that in Greek and Latin that ph was spoken as an f). In Irish lenition a ph would always be spoken as an f. So, aspiration is really not the best term to use.

Basically, the plosive (stop- or explosive) is replaced by its corresponding fricative.

Here is a bit about the causes of lenition.

Transcription and Pronunciation:

The lenited consonant is denoted by a following h.
(In the old script with a dot above the consonant)

pronunciation of lenited consonants at the start of a word:

• A broad gh/dh is a [γ], a "stimmhaftes ch", almost like a (throaty, unrolled) r (e.g. in "Bratpfanne"), slender gh/dh like a German j.
  In my home dialect, Berlin-Brandenburg, a g is often also "lenited", e.g. in "ick sare dir" for "ich sage dir" ("broad gh"), or "Jurke" for "Gurke" ("slender gh"), also similar to the Rhine dialect.
• dh was originally the same as the engl. voiced th [ð] (like in that). Now dh is pronounced more like gh.
• th was originally like the engl. unvoiced th [θ] (as in thing), now spoken more like h. (i.e. there are no "lisp sounds " in Irish anymore )
• sh and th are spoken like h, but, preceding a long a,o,u, a slender sh, th becomes [x'] (like ch in ger. ich): a Sheáin [@ x'a:n'], a thiocfaidh tú [@ x'u:k@ tu:]
• mh partially leads to a nasalisation of the neighbouring vowels, as opposed to bh, other than that, they are the same.
• f and s are indeed fricatives, so fh is ignored completely in speech, sh is spoken as h.

special cases :

d, t, s:

• d, t, s (meist) nicht lenited after d, n, t, l, s! ("dentals"-rule! [ 1 ])
  But attributive adjectives are lenited despite d, t, s
• sc, sp, st, sm, sf  are always unlenited (because then they would be impossible to say)!
• s: in some cases (after an article with words ending in n) instead of lenition: s becomes ts (wenn followed by a vowel or l, n, r)(see t-prefix)

• fh is always mute, this is why a word beginning in fh + vowel (or l, n, r) is treated like a word that starts with a vowel:
  e.g.: m'fhear céile = my husband (not: *mo fhear céile), d'fhoghlaim mé Gaeilge = I learned Irish (not: *fhoghlaim mé Gaeilge), d'fhliuch sé an t-úrlar = he wet the floor
• should a word with f change to drasticly change through lenition, sometimes the lenition is rejected.

• b, m and p are, in part, not lenited after m: e.g. um bosca = around a box
  or in Munster im' instead of i mo: im' poca = in my pocket (instead of: i mo phoca)

• in written Irish, these are not recognizably lenited, but in spoken Irish is there is an audible difference in some dialects: a change from a tensed l, n ([L], [N]) to a relaxed l, n ([l], [n]) e.g.: unlenited: an leabhar [L'aur] > lenited: don leabhar [l'aur]
  By r, this difference ([r]/[R]) is today uncommon.

use:

nouns:

1. after the vocative particle a: a Cháit!
2. feminine nouns in the nominative after the article an (except d, t) in the case of s: s becomes ts (if s precedes a vowel or l,n,r)
3. masculine nouns in the genitive after the article an (except d, t) in the case of s: s becomes ts (if s precedes a vowel or l,n,r)
4. in indefinite genitive attributes after a feminine noun and after the weak plural (but there are many exceptions, more under the section about the genitive)
5. those definite nouns and proper names as a genitive attribute without an article (no matter if the antecedent is masc. or fem.): muintir Cháit = Kate's parents, stáisiún bhus a trí = the Bus #3 stop
6. because it is never possible that 2 genitives come after one another, the first is only lenited and remains in the nominative ("functional genitive")(gender irrespective): e.g. obair bhean an tí = housewife's work, Lá Fheile Pádraig = St.Patrick's Day (lit. "Day of the Feast of Patrick")
7. a noun attribute in compound words (if the 2nd part is not in the genitive): scian phóca = pocket knife
8. a noun in the after numbers 1-6 (but not after 3-6, if the noun that follows is in the plural) after aon also s becomes ts
9. after beirt, dís (2 people), but not after triúr etc.!
10. after an chéad (the first), but not after an dara, an tríú, etc.!
11. after possessive pronouns mo, do, a (a only if it is the 3rd person sg. masc)
12. after a preposition if it lacks an article ar, ó, do, de, faoi, idir, mar, roimh, thrí, thar, gan
    (after ar, idir, gan, see there)
13. after don, den  in the case of s : s becomes ts (in the standard and in Connacht only the fem. nouns)
14. after sa(n) in the standard, Ulster and Munster (always in Connacht, in Munster in the case of f , eclipsis instead)
15. in Ulster always after preposition + article
16. after uile (= everyone)
17. in surnames after Ní, Uí, Mhic, Nic but not after Ó, Mac

verbs:

1. after ní and the preterite verb particle and conjunctions ending in -r: ar, gur, nár, níor, murar, sular, ach ar
2. a verb after a direct relative particle a (except tá, deir): an teach a thógfaidh sé = the house that he wants to build
3. a verb after má (except tá, déir)
4. verbs in the preterite.(except auton. form), imperfect, conditional (actually after the particle do, the particle itself can only still be seen today preceding the vowel and fh): (do) bhí sé = he was, d'fhoghlaim sé = he learned
5. verbal nouns after the preposition a ("to")   tú a phósadh = to marry you

adjectives:

1. after a feminine noun in the nominative: an bhean mhór = the big woman
2. after a masculine noun in the genitive an fhir mhóir = of the big man
3. after a noun in the weak plural that ends in a slender consonant: na fir mhóra = the big men
4. after the prepositions den/don/sa + noun (not after a masculine noun if the noun remains unlenited)
5. directly after 2-6:  e.g. is trí mheasa é = it is three times as bad
6. after 2-19, if with a noun in the singular: e.g.: seacht mbád mhóra = 7 large boats
7. after beirt, if with a noun: e.g.: beirt mhac bheaga = 2 little sons

general:

1. after prefixes:  an-mhaith, fíormhór, seancharr etc.
2. the second part of a compund word
3. words after forms of the copula in the preterite/conditional (base form ba) e.g.: ba mhaith liom = I would like
4. déag after words in the singular ending in a vowel e.g.: seacht hata dhéag = 17 hats
5. déag after weak plural ending in a slender consonant (except cinn): e.g.: trí fir dhéag
6. déag is lenited after dó (but not after trí, sé, naoi)  a dó dhéag = twelve
7. within a word and at the end of a word, due to historical phonetic evolution (pronunciation is then often quite different)
8. some words are mostly lenited (e.g.: dhá = two, cheana = already, chomh = like, chun = to ,
   similarly, also dialectical with dhuit = to you, dhá = to his, if, cheithre = four )

[ 1 ]
The deeper meaning of the "dentals"-rule:
The name of the rule derives from the sounds in question (d,n,t,l,s).
They are not all dentals in the linguistic sense, but they are all coronals, that are produced in the same (homologous) location in the mouth: the tip of the tongue touches the "corona" of the upper front teeth when one speaks d,n,t,l,s.
Because these consonants are produced at the same location, they are easily spoken one after another. On the other hand, the sounds th, sh, dh are produced at different locations. This means that to speak a lenited d,t,s after d,n,t,l,s, would mean extra work for the speaker.
The reason for lenition was originally to simplify speech instead of making it more difficult. This is why after d,n,t,l,s one abstains from lenition of d,t,s.
The same goes for other homologously formed consonants. Also there one could build up "rules" about the groupings (e.g. a "b,m,p rule"), but these hardly play the role that the coronals do.
To generalise, one could formulate the rule so, that it states that homologous consonants cancel out lenition if they occur right after one another.