The five limbs
Panchang means "five limbs": tithi, vaara, nakshatra, yoga and karana. Together they describe the quality of a day.
Tithi is a lunar day — the time the moon takes to gain 12 degrees on the sun. There are 30 in a lunar month, 15 in each paksha. Because a tithi is not the same length as a solar day, a tithi can begin mid-morning and end the next afternoon. The tithi current at sunrise is the one the day is named for.
Nakshatra is the lunar mansion the moon occupies — one of 27 divisions of the ecliptic, each 13°20'. The nakshatra at sunrise governs the day's character.
Vaara is the weekday, each ruled by a graha.
Yoga and karana are derived quantities used for finer judgements.
Rahu Kalam
The daylight period between sunrise and sunset is divided into eight equal parts. One part each day is Rahu Kalam, and its position rotates by weekday: Monday the 2nd part, Tuesday the 7th, Wednesday the 5th, Thursday the 6th, Friday the 4th, Saturday the 3rd, Sunday the 8th. Because sunrise and sunset move through the year, so does the clock time of Rahu Kalam — which is why it must be computed for your city, not read from a fixed table.
New beginnings are avoided during Rahu Kalam. Ongoing work is not affected. Yamaganda and Gulika Kalam are computed the same way and are similarly avoided, though generally treated as less serious.
Abhijit Muhurta
Abhijit is the eighth of fifteen equal divisions of the daylight period — roughly 48 minutes centred on solar noon. It is considered auspicious for almost any undertaking and is the fallback when no other good window is available. It is not observed on Wednesdays.
Why dates differ between calendars
Amanta calendars (Karnataka, Maharashtra, Andhra Pradesh, Tamil Nadu, Gujarat) end the month at Amavasya. Purnimanta calendars (North India) end at Purnima. The same festival can therefore be listed under different month names — the actual day is usually the same.