The best time to come to Seville depends a lot on what you are looking for, and every season has its pros and cons. In general, the best times to visit are in spring and autumn, when the weather is still warm enough for sandals, shorts and t-shirts and sitting outside to eat, but it's not too hot to be comfortably out and about during the afternoon.
Spring - orange blossom and fiestas
Early spring is my favourite time of year to be in Seville. The chill of winter is beginning to dispel, and in late-February and March, the orange blossom, the azahar, is out, and everywhere you go is redolent of its subtle fragrance. A little later are Seville's two big spring festivals, Semana Santa (Holy Week, leading up to Good Friday), and the Feria de Abril, starting two weeks after Easter Sunday. For visitors from Northern Europe and English speaking countries the processions of Easter week, with their hooded penitents and statues of Christ and the Virgin, can come as something of a culture shock. If it's not your cup of tea, it's best avoided, as the crowds make it difficult to get about, the bars and restaurants are all full, and the price of hotels goes sky high.
Summer - hot but less crowded
The obvious disadvantage of coming to Seville in July and August is the heat. And I mean heat. Temperatures can soar to 40ºC or more, so if you are here, find an air-conditioned bar (note that a lot of bars close in August for staff holidays) and grab yourself a cold beer. The upside is that this is low season, and you may well be able to get a good bargain on accommodation. The monuments and other sightseeing spots are generally less crowded too, and by nighttime its cool enough to sit on the rooftop terrace at the Fontecruz or the EME and watch the sun set over the Cathedral with a cocktail or two. There's a perfect moment at nightfall when the sky turns a fabulous deep indigo blue.
Autumn - the season to be out and about
Season of mists and mellow fruitfulness, but without the mists. If you're coming to see the sights and/or for eating in the open air, September and October are the best time to be in Seville. Wander around the city, visit the parks and gardens, take a day trip to Italica, or just sit on the terrace at one of the bars or cafés. Perfect.
Winter - random weather and Christmas lights
It can get surprisingly cold during December and January, particularly in the mornings and evenings, and sometimes wet, too, but there are usually plenty of sunny afternoons for taking a walk around the city. If it's raining you can take in a museum or do some Christmas shopping. The Christmas lights are on throughout December around the Cathedral, in the Plaza Nueva, and on the main shopping streets. Visit one of the traditional belens (nativity scenes) and the belen market, where you can buy your own.