〜と思う
I think (that)
Plain form + と思う · Noun / な-adj + だと思う
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Plain form + と思う says what you think: 明日は晴れると思います, "I think it'll be sunny tomorrow". と marks the content of the thought, just as it marks a quote.
Nouns and な-adjectives need だ: 学生だと思う, 静かだと思う.
Japanese puts the negative inside the thought: 彼は来ないと思う ("I think he won't come"), where English says "I don't think he'll come". 来ると思わない exists but sounds like a strong denial.
と思っています describes an opinion held over time, or someone else's: 母は私が医者になると思っている.
と思います also softens opinions, so you'll hear it far more than English speakers say "I think".
Examples
明日は晴れると思います。
I think it'll be sunny tomorrow.
田中さんは学生だと思います。
I think Mr Tanaka is a student.
この映画は面白いと思う。
I think this film's good.
彼は来ないと思います。
I don't think he's coming.
日本語はそんなに難しくないと思います。
I don't think Japanese is all that hard.