Use anti-plagiarism software. Teachers can use an anti-plagiarism service such as turnitin.com to see if students have copied and pasted from the Internet or bought a pre-existing paper. Students upload their papers to the service and it checks the content against its database, which includes websites, academic journals and previously-uploaded papers, for duplicate content. It won't catch all plagiarists, but can quickly weed out the most obvious.
Look for changes in writing style and word choice. If a student has copied and pasted a section of their paper without properly citing it, it will often be obvious due to the change in writing style and word choice. The paper will suddenly go from sounding like a college freshman to sounding like a professional writer.
Look for discrepancies with other student work. If a student pays a friend to write their paper or buys a custom paper from a paper mill, Steps 1 and 2 won't catch it. The easiest way to spot this type of plagiarism is to look for discrepancies between the paper and other student work. If a student fails a multiple choice midterm but then turns in an amazingly well-written paper in the same class, this may be a sign of plagiarism.
Talk to the student. If you suspect plagiarism but can't prove it definitively, talk to the student about their paper. Plagiarists will often give themselves away by not being able to talk about what they have supposedly written.