Seville is in the west of Andalusia, Southern Spain, about 100 km from the coast. Below are details of how to get from Seville city center to Seville's airport.
Buses to and From Seville Airport
With a large number of flights arriving in Seville (or Sevilla in Spanish), there is a regular bus to and from the airport. Unfortunately, the old stop in the center of Seville (at Puerta de Jerez) has been removed from the route and now the most central bus stop is near the Prado de San Sebastian bus station, though it does pass by the main train station (San Justa). Note that there are two bus stations in Seville and San Sebastian is the one furthest from downtown (the bus station in the center is called Plaza de Armas - read more about Seville Bus and Train Stations).
Taxis to the Bus Stations and Airport
If you are planning to get a taxi to the bus station to catch a bus to the airport, it makes sense to miss out the bus station completely and take a taxi straight to the airport, especially if you are traveling in a group (taxis in the center can be delayed heavily by traffic - you may end up paying 10€ just to get to the bus station, whereas a taxi straight to the airport has a flat fee of under 20€).
How to Get from Seville to Other Destinations in Spain
Seville's main train station is called San Justa and is well connected - a train to Madrid takes just 2.5 hours. The bus from the airport stops outside it. Travel by bus is a little more complicated - there are two bus stations in Seville and the one you arrive at from the airport (Prado de San Sebastian) is unlikely to be the one you want (most buses leave from Plaza de Armas) and you'll need a taxi to get between the two. If you are not planning on staying in Seville but don't know which bus station you want, get a taxi from the airport and ask your taxi driver which station you need (he should have rudimentary English skills).
Alternatively, get the lowdown before you leave: Seville Bus and Train Stations.