An exchange is a group of telephone voice line circuits grouped together by a single digital access cross-connecting switch or set of switches. This set of circuits comprise the smallest unit of connections that can be identified by the North American Numbering Plan. Calls between phones in an exchange are said to be 'local calls' and are usually billed at the lowest rate offered as part of the subscriber's current calling plan.
Your telephone is connected via the local loop to a single central office. That central office and all the other phones connected to it are referred to as your local exchange.
Everyone in your local calling area was connected to your Local Exchange. Calls from your local exchange to any other exchange were connected across tie lines (later called trunk lines) to a 'foreign exchange'.