Almost every operator will tell you June to October. They're not wrong, but they're answering a narrower question than the one most people are really asking.
The dry months, roughly June to October
The grass is short, the water is scarce, and animals gather at whatever's left of it. Visibility is at its best, and the roads are at their easiest. It's also the busiest and most expensive time, and in the Ngorongoro Crater in August, you definitely won't be alone.
The green months, roughly November to May
There are two rainy spells inside this stretch: the short rains around November and the long rains in April and May. In between, especially in January and February, the southern Serengeti plains are green, the calving is in full swing, and the predator action that follows is the best of the year. You'll get fewer vehicles, lower prices, better light and more dramatic skies.
The trade-off is real, though. April and May can be properly wet, some camps close, and black-cotton soil turns into a real problem. When a month is a gamble, we'll tell you so instead of selling it to you as a bargain.
If the migration is the whole point
Then the month decides where you stay, not whether you go. The herds are in the ecosystem all year. Have a look at the migration calendar to see where they usually are, and please do read the warning on it.
Month by month
We've got a page for every month: when to visit Tanzania, with the weather, the crowds and where the migration is for all twelve.