There are various ways to count music, from simple numbers to counting syllables to beat placement syllables. Here are a few.
Ta Titi system:
This system allows the value of each note to be clearly represented no matter its placement within the beat/measure. The folk song lyric "This Old Man, he played one, he played knick-knack on my thumb, with a knick-knack paddy whack, give my dog a bone, this old man came rolling home" would be said, "titi ta titi ta titi titi titi ta-tiri titi tiriti tiritiri ta titi titi titi ta"
du / du-deh / du-ta-de-ta system! Need more info.
Down-ee up-ee – beats are down, up-beats are up, subdivisions are “ee” but… need more info!
The folksong lyric "This Old Man, he played one, he played knick-knack on my thumb, with a knick-knack paddy whack, give my dog a bone, this old man came rolling home" would be said, "down up down down up down down up down up down up down up-ee down up down-ee-up down-ee-up-ee down down up down up down up down."
Takadimi system: The beat is always called ta. In simple meters, the division and subdivision are always ta-di and ta-ka-di-mi. Any note value can be the beat, depending on the time signature. In compound meters (wherein the beat is generally notated with dotted notes), the division and subdivision are always ta-ki-da and ta-va-ki-di-da-ma.
The note value does not receive a particular name; the note’s position within the beat gets the name. This system allows children to internalize a steady beat and to naturally discover the subdivisions of beat, similar to the down-ee-up-ee system.
The folksong lyric "This Old Man, he played one, he played knick-knack on my thumb, with a knick-knack paddy whack, give my dog a bone, this old man came rolling home" would be said, "tadi ta tadi ta tadi tadi tadi tadimi tadi takadi takadimi ta tadi tadi tadi ta."
Ultimately, musicians count using numbers, “ands” and vowel sounds. Downbeats within a measure are called 1, 2, 3… Upbeats are represented with a plus sign and are called “and” (i.e. 1 + 2 +), and further subdivisions receive the sounds “ee” and “uh” (i.e. 1 e + a 2 e + a). Musicians do not agree on what to call triplets: some simply say the word triplet (“trip-a-let”), or another three-syllable word (like pineapple or elephant) with an antepenultimate accent. Some use numbers along with the word triplet (i.e. “1-trip-let”). Still others have devised sounds like “ah-lee” or “la-li” added after the number (i.e. 1-la-li, 2-la-li).
The folksong lyric "This Old Man, he played one, he played knick-knack on my thumb, with a knick-knack paddy whack, give my dog a bone, this old man came rolling home" in 2/4 time would be said, "one and two one and two one and two and one and two and uh one and two ee and one ee and uh two one and two and one and two."
http://www.takadimi.net http://faculty.bush.edu/mark.francis/curriculumrhythm.pdf