As I understand your question, you're asking what would happen if the Earth stopped rotating on its axis, not what if it stopped revolving around the Sun--is that right? Ignoring the effects of what would happen from, say, suddenly stopping the Earth's rotation... Ironically, what you're describing, with one side always facing the Sun, isn't "not rotating" at all. If the Earth stopped rotating on its axis, then we would still have day and night--but a full daily cycle would be the same length as the year. Our day would match our year. Instead, what you're asking about is what if Earth were tidally locked to the Sun, the way Mercury is, and the way the Moon is to us. We're actually headed that direction...very very slowly. Eventually, the Earth will be tidally locked to the Sun, if it survives long enough for it to happen (I forget if that takes longer than the 4.5 billion years before the Sun turns to a red giant). Anyhow, if the Earth stopped rotating, the days would get very hot and the nights would get very cold. Think about how it gets around 2-3pm in the summer--but now the Sun isn't going away and it keeps getting hotter for the next several months. Likewise think how cool it gets in the early morning hours--now there won't be light and warmth for some months, and it'll keep getting colder. This would also $#@! real badly with plants' metabolic cycle and probably cause most of them to die out, which would make the heat and cold the least of our worries. On the other hand, if the Earth became tidally locked to the Sun, it'd be even worse. We'd never reach Mercury-like extremes, though, because of the insulation and convection properties of our atmosphere--but it would become quite literally an unlivable place for us.