The five daily prayers
Allah made five prayers a duty at set times. The Quran says: 'Indeed, prayer has been decreed upon the believers a decree of specified times' (Surah an-Nisa 4:103). Their times follow the sun: Fajr at true dawn, Dhuhr just after the sun passes its peak, Asr in the late afternoon, Maghrib at sunset, and Isha once twilight is gone. Sunrise is shown too - it marks the end of Fajr's time, and you do not pray then.
Why methods give slightly different times
Dhuhr, Asr, and Maghrib follow the visible position of the sun, so they vary little between methods. Fajr and Isha depend on how far the sun is below the horizon - the twilight angle - which trustworthy bodies have measured a little differently (the Muslim World League, Umm al-Qura in Makkah, ISNA in North America, Karachi, and others). Choose the method your local mosque follows; the differences are usually only a few minutes.
The two views on Asr
For Asr there are two well-known opinions. The majority (Maliki, Shafi'i, Hanbali) hold that it begins when an object's shadow - beyond its shadow at midday - equals the object's own length; the Hanafi school holds that it begins when that shadow is twice the length, which makes Asr a little later. Both are valid; pick the one you follow.