Lebanon has a Mediterranean climate on the coast and an alpine climate in the mountains, which means the best time to visit depends on what you want to do.
Spring (March-May) is the sweet spot for most travelers. Temperatures are mild and pleasant (18-25°C on the coast), wildflowers carpet the mountains, and it is possible to ski in the morning and swim in the afternoon during early spring. The country is green and beautiful.
Summer (June-September) is hot and dry on the coast, with Beirut reaching 30°C or more and high humidity. This is peak season -- Lebanese expats return, beach clubs are packed, and the mountain villages fill up with people escaping the coastal heat. Summer is festival season, with events at Baalbek, Beiteddine and Byblos drawing international performers.
Autumn (October-November) is another excellent time to visit. Temperatures cool down, crowds thin out, and the light is gorgeous. Occasional thunderstorms arrive but rarely last long.
Winter (December-March) brings rain to the coast and heavy snow to the mountains. Beirut averages about 10°C in January -- cool and wet but rarely freezing. The ski resorts in the mountains (Mzaar Kfardebian, the Cedars) operate from December through March or April, making Lebanon one of the few places where you can ski within a short drive of a Mediterranean beach.
Rainfall is concentrated between November and March. Summers are almost completely dry. The Bekaa Valley is drier and more extreme in temperature than the coast.