In 1538 Thomas Cromwell, the Vicar General to King Henry VIII, declared that all marriages, baptisms and burials should be recorded; normally these are kept at the Parish Church, and are known as 'Parish Records'. In 1598 an Act was also passed that meant these records should be copied and sent to the bishop, these are known as Bishop's Transcripts.
Although most parish registers usually date to around the mid-16th Century, when Elizabeth I declared that accurate registers should be maintained, some are also available from 1538. Elizabeth I had instructed that they should be preserved in bound books and not on any old scrap of writing material. Even after her efforts, many registers have been lost over the years and those surviving today are now usually safely deposited in the local record office, not the church they are associated with.
Parish registers can contain large gaps, such as for the Commonwealth period, which can leave the mid-1600s looking sparse. You also get volumes which have gone missing over the years leaving frustratingly large gaps. Small gaps are not as bad unless they happen to coincide with the event you are looking for. These are usually due to negligence, often occurring where the clerk didn't enter them at the time and forgot to do it later. If you are lucky, some of the larger gaps in Parish Records caused by lost volumes may be covered by the Bishop's or Archdeacon's Transcripts. These are copies of the events copied from the Parish registers which were sent to the Diocese each year. BT's may themselves be incomplete, especially at the beginning or end of the periods.
Unlike the Census or the Civil registration material, Parish Records are scattered all across the country, so it will be well into the future, if ever, that we will be able to search them online in the same way. Most churches have deposited records over 100 years old at their local record office, but some still retain burial registers for example, that were started in the last century and still are not full yet. Chapel records are not as well preserved, some are in private hands and many have been lost. However, there are many parish register transcripts published by various parish record societies, historical groups and individuals and these lend themselves to being made accessible online.