'Brexit' is shorthand for a Br(itish) exit from the EU. The precedent was set by the Greek crisis, where 'Grexit' was used to describe the possible exit of Greece from the euro currency and the EU as a whole.
On Thursday 23 June, there will be a referendum on whether the UK should remain in the European Union (EU). The voting options will be "Remain a member of the European Union" or "Leave the European Union". Should the latter gain more votes, this would lead to 'Brexit'.
With so much reporting in the media claiming all sorts of implications for staying or leaving the EU, see below a link to a website that provides more information regarding the various myths that are being discussed frequently, such as the impact on immigration and the economy.