Before they converted to Christianity and adopted the Roman calendar, the early English ('Anglo-Saxons') used a calendar based on the cycles of the sun and the moon.
The year was a solar year, and the two most important dates were the summer solstice (Midsummer, the longest day of the year) and the winter solstice (Midwinter, the shortest day of the year). The winter solstice was called Guili, or Yule, and is the origin of our word "Yuletide" for Christmas - for more details, see the page on Yule. Each new year began at Yule.
The year was divided into two seasons, governed by the spring and autumn equinoxes (the points when the day and night are of exactly equal length). The season when the days were longer than the nights was called summer, the season when the nights were longer than the days was called winter.
Months were reckoned by a full cycle of the moon. Since Bede tells us that winter began at the full moon of October, the months presumably also began at the full moon. The number of days in a solar year isn't an exact multiple of the number of days in a lunar cycle, so there are 12-and-a-bit lunar months in a year. As a result, the English months moved around in relation to the solar year. Every so often an extra month was added at Midsummer, making a 13-month year, to keep the months aligned roughly with the seasons.
We know this from a contemporary document, Bede's On the Reckoning of Time, written in 725 AD. Bede was concerned mainly with teaching his students how to calculate Christian festivals, such as that perennially knotty problem of the early Church, the correct date of Easter. Fortunately for the scholar of early England, however, Bede kindly added a chapter (Chapter 15) explaining how his people had calculated months before they adopted Christianity. It provides the main documentary evidence we have for the pre-Christian English calendar.
Bede: The Reckoning of Time. Translated by Faith Wallis. Liverpool University Press, 1999, ISBN 0-85323-693-3.