I'll try to explain what little I know:
Ashtami is called a tithi and is the 8th day from either a full moon (Purnima) or a new moon day (Amavasya). The waxing fortnight (between new moon and full moon) is called Shukla paksha and the waning fortnight is called Krishna paksha. The days are counted from new moon or full moon as Prathama, Dvitiya and so on. It goes up to chaturdashi (14th day)
Bhadrapad is the name of the 6th month of the calendar and typically extends between August and September. The year starts with Chaitra, Vaishakha, etc.
Shraavana refers to the star (nakshatra) and is the 22nd of 27 stars. The sequence begin with Ashvini, Bharani and so on. This is the constellation in which the moon is when a person is born.
When we quote our birth date, we note the month and the star, and traditionally we celebrate our birthdays on the same combination of month and star.
As the wikipedia states, Krishna's birth is supposed to have happened on the 8th day of Bhadrapada month under the star Shraavana. There are regional differences on exactly when to celebrate.
This is overly simplified and there are numerous ifs, whens and buts. It also makes a huge difference if you're primarily following a solar calendar which needs to be overlaid on the lunisolar calendar.
I hope somehow knowledgeable can fill in and also correct any misconceptions that I may have presented here.