Spring (March-May) is the best time to visit. Temperatures are warm but not brutal, azaleas and dogwoods are blooming, and the spring bird migration on Dauphin Island is spectacular. The downside is severe weather -- Alabama sits in tornado alley, and spring storms can be intense.
Fall (September-November) is equally pleasant. The humidity drops, the leaves turn in the northern mountains, and college football season brings the state to life. October and November are ideal for hiking and outdoor activities.
Summer (June-August) is hot and sticky. Temperatures in the 90s with high humidity are the norm. The Gulf Coast beaches are at their busiest. If you can handle the heat, you will find fewer crowds at inland attractions.
Winter (December-February) is mild by northern standards but cool enough to be comfortable for sightseeing. Snow is rare except in the mountains. Mardi Gras in Mobile, which predates the New Orleans celebration, falls in February or March and is worth timing a visit around.