There are three main airports in Portugal with commercial services:
Porto - in the North - Francisco Sá Carneiro Aiport
Lisbon - in the centre and Capital City -
Faro - in the South and the airport for the Algarve region.
Low cost airlines operate from all three to various European destinations:
For the UK and Ireland see among others:
Easyjet.com
Ryanair.com
Monarchfly.com
For Germany:
Airberlin.com
Tuifly.com
Germanwings.com
For Spain:
Vueling.com
For a full list of operators from Faro Airport see:
http://www.solpraiaferias.com/index.php?content=cheg_en
For the other airports:
http://www.ana.pt
There are three daily flights between Lisbon and the Algarve by the national carrier - Air Portugal / TAP. I do not recommend it as it is costly, and flights are often delayed /cancelled.
There are excellent train and bus services between most centres in the Algarve and Lisbon.
