In the rhyme quoted:
Tid, Mid, Misera,
Carl, Paum, Good Pose-Day
the Carl, Paum and Good Pose-Day refer to the 5th Sunday in Lent (14 days before Easter), the 6th Sunday in lent(7 days before Easter) and Easter Day itself.
Carl Sunday or Carling Sunday is more commonly called Passion Sunday, the first day of the two-week season of Passiontide. It is traditional in Northern England to eat cakes made out of brown peas, called carlings. Although many say the food gave the name to the day (rather like Pancake Tuesday), others (including Rev Hudson Baker quoted by @Marc) say it was the other way round. Carl comes from the Old English word cearu meaning suffering, sorrow and grief, relating to the beginning of the Commemoration of the Lord's Passion on that day. This is possibly also related to the German name for Good Friday, karfreitag.
The day after Carling Sunday is known to schoolchildren as Farting Monday, due to the effects of the peas.
Paum, or Palm, Sunday is known throughout the Christian world as the commemoration of the Triumphal Entry into Jerusalem during which palm branches were waved.
Good Pose-Day, is in some versions of the rhyme Paste-Egg Day and in some Pace-Egg Day. In all cases it is probably a form of pascha, from where the word paschal, pertaining to Easter, also comes. Paste-Egg relates to the custom of colouring eggs, and Pace-Egg relates to the custom of rolling eggs, both Easter customs.
The first line: Tid, Mid, Misera is very obscure. They are usually regarded as the second, third and fourth Sundays in Lent respectively. This is said in the Gentleman's Magazine of 1788, quoted here. The origins of these names are obscure. Reverend Brady in 1812 said some suggested they related to the old pre-Reformation hymns sung on particular days. The Ambrosian hymn "Te Deum" sung on the second Sunday of Lent, another hymn "Mi Deus" on the third, and Misera Mei (Psalm 51) on the fourth. However the scholar J M Neale suggested Misera was a corruption of Reminiscere Sunday, which as Ken points out is a name for the Second Sunday in Lent. He was an expert on pre-Reformation English liturgy but knew of nothing that could explain the "Tid Mid Misera" line. The Te Deum Laudamus theory is apparently unlikely because the Te Deum was sung all year round except in Advent and Lent, so hardly likely to give its name to any Sunday, least of all one in Lent.
Baker suggests that Tid referred originally to the Second Sunday, it occurring after ten days of fasting, and tid being related to tithe or ten. Mid and Misera were, according to him one name referring to the fourth Sunday and meaning mid-Lent.
Another explanation is that they may be obsolete counting words. Misera, the fourth Sunday in Lent, bears a slight similarity to methera, a Cumbrian word for fourth.
There is no commonly-accepted derivation for tid mid misera.
The fifth Sunday before Christmas, the Sunday before Advent, is known as Stir-up Sunday, partly because it is the traditional day to stir Christmas Puddings and partly because the Collect for that Sunday begins "Stir up, O Lord, the hearts of thy faithful people ...". The Collect reminding people of the need to stir the puddings.