Apparently I can answer but not comment with such a new account. This is really more of a comment to Daniel R. Collins' great answer.
Anyway, in my country, there's a concept of the "summer holidays" when kids otherwise going to school are expected to relax and/or do other non-school-related activities (such as summer camps - note that "winter camps", if any, are much rarer). This is basically June to August (July to August for students, whose classes end later).
As far as I understand it, these holidays are in the summer (as opposed to winter) because the weather in summer is much more pleasant, and thus much more appropriate for relaxationary activities (such as country vacations, or traveling). And, yes, historically (and partly to this day), there's some farming-related stuff too (for the kids to help with at some of the summer camps; this is not much the case today, except in the especially rural areas, but was very popular as recently as several decades ago).
Incidentally - something that did not come up in Daniel R. Collins' answer, or indeed, as I write it, anywhere eise in the comments - the calendar year does not necessarily start in January; in fact this seems to be a (historically speaking) rather new tradition even where it does. (I do not know where it originates; it might have had to do with Christmas.)
In particular, the Hebrew (and therefore Israeli) calendar year starts in September, not much after the academic year. The Muslim calendar year doesn't really start at any seasonally defined point at all; except in Iran and Afghanistan, where it starts in March.
More historically, the British calendar year started in March prior to 1752 (the financial year still does, though it's been moved to April by the change to Gregorian). The Russian calendar year started in September during the 16th and 17th centuries (and in March before that). Of course, as mentioned by vonbrand, the old Roman year started in March (from which the names of our autumn months originate).
Other traditions have yet other starting dates for the calendar year; I've heard somewhere that for every day of the year there's a tradition somewhere that puts the New Year on that day. That much is probably not true, but there are definitely very many. Not many of them, save for the modern Gregorian one, are in January (the traditional Chinese new year almost is, but it's more often in early February); December seems slightly more common (mainly due to the winter solstice).