शिरोदय = शिर + उदय = Head + Rising :

Those Rashis which rise through their head part while rising are Sheerodaya Rashi. Head and hind part of Rashi can be separated because Rashis are specific constellation having specific pattern of animals or object. i.e. When they rise at east when head part is seen first while rising then they are called Sheerodaya Rashi.

Sheerodaya Rashi are: Mithun (Gemini), Simha (Leo), Kanya (Virgo), Tula (Libra), Vrishchik (Scorpio), Kumbha (Aquarius).

पृष्ठोदय = पृष्ठ + उदय = Hind + Rising :

Those Rashi which rise through their back part while rising are called Pristhodaya Rashi.

Pristhodaya Rashi are: Mesha (Aries), Vrish (Taurus), Karka (Cancer), Dhanu (Saggitarus), Makara (Capricorn).

उभयोदय = उभय + उदय = Both + Rising

Those Rashis which rise from their both head and hind part are called Ubhayodaya Rashi. Pisces is the Ubhayodaya Rashi.

Application:

The application of Sheersodaya, Pristhodaya and Ubhayodaya Rashi are useful in finding Rashi Bala (Power of Rashi) and Dasha Bala.

Rashi Bala: Ubahayodaya Rashi are powerful at both day and night.

Pristhodaya Rashi along with Gemini are powerful at night.

Sheershodaya Rashi: Powerful at day.

The above method to determine Rashi bala (potency of Rashi) is only one of the many methods to determine Rashi Bala.

Dasha Bhukti (Dasha Period): During the period of planets i.e. Dasha if the planet is situated at Sheerodaya Rashi then it gives more fruits especially in first half of Dasha, if Pristhodaya then it gives more fruits especially in last half of Dasha, if Ubhayodaya gives fruit throughout the Dasha.

But again this is only one of the aspect among many other aspects to determine results of Dasha Bhukti Period.