What is audax?

Audax is long distance cycling.
Audax events are organised by CTC groups, cycle clubs and sometimes by individuals under the rules of AudaxUK - all are volunteers.  The routes, all over the UK and usually in multiples of 100km in length up to 600km, are devised by cyclists with good local knowledge, so you can be assured of a route using minor roads wherever possible and offering the best riding in the area.
The events are open to all cyclists for a modest fee (generally less than £10).  Entrants who are not full members of AUK or CTC (which provide third party insurance cover) are required to pay £2 extra for temporary AUK membership.  Entrants resident overseas will have to arrange their own insurance and be prepared to show proof to the organiser.
While there is a great deal of cameraderie on the rides, particularly the longer rides, the emphasis is on self-sufficiency.  There is no signing of the route (you get a detailed paper routesheet in advance of the event, and maybe a gpx track), no mechanical support and no broom wagon; if you cannot complete the ride you will be expected to make your own way back.  Numbers of participants tend to be lower than for cyclosportives - typically tens instead of hundreds.  Any sort of bike can be used, provided it is comfortable and roadworthy.  While drop-handlebar road bikes are by far the most numerous, participants are likely to see recumbents, tandems, folders, utility, even vintage, bikes on audax events.  Participants used to be required to have mudguards fitted on their bikes, but in most cases these are no longer required.  And there are no rules regarding helmets.
Audax rides are not races*.  Indeed while the organiser will make a note of riders' finishing times these are not published.  This is possibly the biggest differentiating factor between audax and cyclosportives.  Riders are required to complete the events at a minimum speed of 12 - 15 kph (around 8 - 9 mph) depending on the length of the event, but at a maximum of 30 kph (18.75 mph).  These are average speeds over the event, and include any stoppage time for eating, or sleeping on the longer events.  Note that for some events (designated "BRM" in the AUK calendar listings) the maximum time for completion of the event is based on the nominal (200, 300 km etc) rather than the actual event distance, which can mean slightly higher effective minimum speeds are required.  Riders all set off at the same time, and then have to visit a number of control points to collect proof of passage.  These controls are often in cafes, where there may be a "controller" stamping the brevet card that is issued at the start.  Other forms of control can require collecting a receipt, or answering a question printed in the brevet card (info controls).  At the end of the event the rider signs the brevet card and leaves it with the organiser.  This is then validated, and will be returned to the rider as a permanent record of the event.  Some events now return the brevet card to the rider at the end of the ride, once the detail has been recorded.
The flagship event within the AudaxUK calendar is the London-Edinburgh-London event, of approx 1400km.  This is held every four years, with the next LEL in July 2017.  There are other >1000km events in other countries, the most famous being the 1200km Paris-Brest-Paris, also held every four years (next in 2019).
A full list of audax events in the UK, and a selection of FAQs, can be found on the Audax UK website (use the drop downs at the top of that page to refine your search, and to see events more than 4 months ahead).
To enter an audax event you should go to the AudaxUK website linked above, identify the ride you wish to do, and follow the entry instructions there.  For local events the page Events in the Thames Valley area on this site provides links to the individual event pages.  Many organisers encourage electronic entry via PayPal; you do not need to have a PayPal account to use this.  Note that with paper entries (which can be used for any event) it is necessary to send, with the entry form and cheque, 2 SAEs.  These are for the routesheet to be sent to you before the event, and the brevet card to be returned after the event.  Electronic entries sometimes carry a surcharge to cover the cost of envelopes and stamps.
It is often (but not always) possible to enter an audax "on the line".  You will still be required to complete the entry form and provide one SAE, and there may be a surcharge on the normal cost.  On-the-line entrants may have to wait for the organiser to deal with those who have entered in advance, and so may be a few minutes late starting the event.  Increasingly organisers are charging extra for on-the-day entries to reflect the extra work they create at the start of the event.
* While audax events are not races, that is not to say there is no competitive spirit.  For many this is just riders competing against themselves (a new personal best, perhaps) or the elements.  Others are more competitive, and you will often hear talk of "being first back" or somesuch.  It is sometimes said that sportives are for riders pretending to race, while audax is for riders pretending not to race.  There is, however, no pressure except the overall time limit.