Manner of articulation

You can classify consonants by how they are produced. While the air passes through your mouth, you close it and create friction, thus adding some noise to the signal. The way you do that determines the manner of articulation, and I'll list it roughly by how far closed the mouth is.

Plosive/Stop
An explosive sound, created by a sudden obstacle in your mouth. You can do that with the lips (p, b), with the tip of the tongue (t, d), or with the back of the tongue (k, g).

Nasal
Create a prolonged obstacle in your mouth, while letting air flow into the nasal cavity. Again, you can do that with the lips (m), with the tip of the tongue (n), or with the back of the tongue (the n in finger).

Fricative
The mouth is a bit more open, and you create a sound that's a bit like white noise. Examples: f, s, sh, h.

Affricate
A plosive that resolves into a fricative. Examples: j, ch in church

Vibrant
Like the name says, these sound like a vibration. Example: r. There are different subclasses for vibrants, mainly having to do with their length.

Tap/Flap: The tip of the tongue touches the palate or the teeth just once. You'll find these in Gaelic, the IPA symbols are [ɾ] and [ɾʲ], respectively.

Trill: A long, rolled r. Gaelic has one too, created with the tip of the tongue against the gum ridge. The IPA symbol for this sound is [r].

Approximant
The most open type of consonant. Again, there are two subforms:

Semivowel: Closed versions of corresponding vowels, very often of i and u. English has these too, the orthographic symbol for the second one is w.

Lateral: The sound passes by one or both sides of the tongue. Example: l.

Place of articulation

The second criterium for classifying consonants is where you produce them. Let's take it from front to back, again with examples from English.

Bilabial
Use both lips to produce the sound. Examples: p, b, m.

Labio-Dental
The lower lip touches the upper teeth. Example: f.

(Apico- ) Dental
The tip (apex) of the tongue touches the upper teeth. Example: th.

(Apico-) Alveolar
The tip of the tongue touches the gum ridge (alveoles). Examples: t, d, n, s.

(Dorso-) Palatal
The back of the tongue touches the roof of the mouth (hard palate). Examples: sh, k in keep.

(Dorso-) Velar
The back of the tongue touches the roof of the mouth, but farther back where it becomes soft (soft palate, velum = "sail"). Examples: c in can, n in finger.

Uvular
Where your mouth becomes the throat. The uvula is the tiny bit that hangs down and that you can use to close off the nasal cavities, e.g. when you hold your breath. Gaelic doesn't produce any sounds there, so we need not be concerned with those.

Pharyngeal
The Pharynx is the part of your throat above the larynx (voicebox). Example: h.

Laryngeal
The larynx is that knob in your throat where the vocal chords are. Also known as Adam's Apple for the male part of the species. Example: The so-called glottal stop. English has no orthographic symbol for it as such, but you can hear it in words that begin with a vowel, where it is added to the beginning before you produce the vowel. In some dialects, it's the tt in words like better.

Voiced vs. Voiceless

Voiced consonants are consonants where your vocal chords swing. Examples: b, n, j, r, l.

Voiceless consonants are those where the vocal chords do not swing. Examples: p, s, k.

You can test easily whether a consonant is voiced. Place two fingers on your throat and say "ssss". You don't feel a vibration, so the sound is voiceless. Then say "zzzz", and you will feel a vibration, so the sound is voiced.


Aspiration

In English, when a word starts with a p, you add a little pharyngeal fricative to the end. You guessed it, it's an h. But it's not a full h, just a little one. So, you represent it in the IPA with a [pʰ]. The romance languages don't do this, so that's one way to get an accent! Gaelic has both sounds, [pʰ] and [p], so it's important to know the difference. It also uses preaspiration, where a little fricative is added before the plosive instead of after it.


Secondary articulation

Now if this wouldn't be complicated enough, you can introduce a second manner or place of articulation to a consonant, in addition to the main one. I'll only introduce two examples here.

Palatalization
In Gaelic, you'll come across variants of dental stops that have a palatal component. Example: Gaelic [t̪ʲ]. Sounds a bit like the ch in church, but the tip of the tongue touches the teeth.

Velarization
The back of the tongue is raised. Example: the l in English call. IPA Symbol: [lˠ]. Languages that don't do that are German and French, which again makes for an accent.

Here's the classification for the vowels that you'll need for Gaelic. You can also check out the Wikipedia page on vowels.

Height
This determines how open or closed your mouth is. [ u] is a closed vowel, [a] is an open vowel.

Backness
Is the vowel pronounced more at the front of the mouth or rather towards the back? [ i] and [e] are front vowels, [a] is a middle vowel, [o] and [ u] are back vowels.

Roundedness
Are your lips round or straight? [o] and [ u] are round vowels, [a], [e] and [ i] are unrounded vowels. Note that this feature is present in English whenever you have a non-back vowel. However, this is not so in Gaelic.

Nasalization
English only has nasal consonants, but Gaelic also has nasalized vowels. You're probably familiar with the phenomenon from French, where the -on in baton is the nasal vowel [õ]. Don't worry if you can't produce these at first, you won't be saying something entirely different from what you intended by accident. Gaelic Example:

bha (past tense of the verb bi): ['va]
mhath (lenited version of the adjective math, we'll get to lenition later): ['vãh]

Note how the first [v] is spelled bh and the second one is spelled mh, this signals the nasalized vowel.

Length
Vowels can be long or short. Example: eat vs. it. Vowel length is an important feature of Gaelic and is usually marked with those pesky accents you have trouble getting on your keyboard. Speakers of romance languages will have a bit of trouble with this at the beginning, because their languages don't distinguish long and short vowels.

You can listen to Gaelic vowels at akerbeltz.


Stressed vs. Unstressed vowels
Unstressed vowels are usually reduced, so you have less of them than of the stressed set. They are often produced pretty neutrally, neither open nor closed and neither front nor back. If you hear anybody talking about a schwa or a schwi, they mean one of these reduced vowels. Gaelic has three unstressed vowels: the schwi [ɪ] is pronounced a bit towards the front, the schwa [ə] is a middle vowel, and [a] is pronounced more openly. You can listen to Gaelic unstressed vowels at akerbeltz.

Diphthongs (from Greek: "with two sounds/tones") are made up of a vowel and a semivowel. They are one sound really, but for practical learning purposes just treat them as two sounds. Falling diphthongs start with the vowel element, like English bow (['baw] or ['bow], depending on what you mean). Rising diphthongs end with the vowel element, like [ja] in English yard. You can listen to Gaelic diphthongs at akerbeltz.

Some English vowels always turn into diphthongs when they are stressed (compare the o in "bow" (['bow]) to the o in "more", the first one is a diphthong, the second one is not), but this does not happen in Gaelic.