Turkey is large enough that the best time depends heavily on where you're going.
Spring (April-May) is the best overall time for most of Turkey. Temperatures are warm but not brutal, wildflowers are out across the Anatolian plateau, and tourist crowds — while growing — haven't yet peaked. Good for Istanbul, Cappadocia, the Aegean coast, and the historic sites. Accommodation is more expensive than winter but easier to find than summer.
Summer (June-August) is peak season on the Mediterranean and Aegean coasts, with warm sea temperatures and long days. It's the right time if you want to swim and lie on a beach. But the same period is punishing for sightseeing in Cappadocia (scorching), eastern Turkey (very hot), and Istanbul (crowded and humid). July and August see the highest prices, the most tourists, and the most domestic Turkish holiday traffic — book well ahead.
Autumn (September-October) is arguably the sweetest time to travel. The sea is still warm from summer, the crowds thin, the light turns golden, and temperatures are ideal for walking and sightseeing. October is particularly good — one of the better months to be in Istanbul. Wine harvest in the Aegean wine regions (Bozcaada, Urla) also happens in September.
Winter (November-March) is low season on the coasts, and many hotels and restaurants in resort towns close entirely. But winter is a fine time for Istanbul — there are far fewer tourists, prices drop, and the city's tea houses, mosques, and covered bazaars are at their most atmospheric. Cappadocia looks extraordinary with snow. Skiing is possible at Uludag near Bursa and at Erciyes near Kayseri from December to March. Eastern Turkey is harsh in winter — roads close, temperatures drop severely.
Ramadan varies by year (it moves forward about 11 days annually). During Ramadan, some restaurants close during the day and the country takes on a different mood — more introspective during the day, livelier at night after the fast breaks (iftar). It's not a reason to avoid Turkey, but it's worth knowing about. Seker Bayrami (end of Ramadan) and Kurban Bayrami (Feast of Sacrifice) are the two major holidays when domestic travel peaks, transport fills up, and many businesses close for 3-5 days.