Plagiarism is distinct from copyright in several ways, though the two are related concepts, as both involve intellectual property. How are they different?
Copying
Plagiarism is concerned with ideas as well as expression. Unattributed ideas are as much a concern to plagiarism and uncredited expression. Copyright, meanwhile is only concerned with expression of those ideas, and protects that expression only if it is original and minimally creative.
Plagiarism is concerned with any amount of unattributed material, large or small. Copyright tends to be only concerned with copying that is more than "de minimis." It needs to be more than a trivial amount. And quantity can have a direct bearing when a court is examining possible copyright infringement, as in the case of fair use.
Plagiarism is concerned with unattributed ideas and expression from any time period, even ancient times. Copyright is only concerned with a defined copyright term (number of years). Anything that falls outside of this term of copyright protection is considered to be in the public domain, and thus unprotected.
Attribution
Plagiarism is all about lack of attribution--a failure to credit the author. Copyright is unconcerned with attribution, but rather is concerned with the use of the material without the proper license or permission, even if the use is credited to the original author.
Intent
Plagiarism is generally tied to intent. If a person can explain that they inadvertently copied from another source, then charges of plagiarism are often dropped. Copyright on the other hand is entirely unconcerned with intent, except in criminal cases. Copyright is concerned only with the potentially infringing use of the material in question, even if the act is subconscious or otherwise not consciously intended.
The harm caused by plagiarism can be diffuse and societal. Society has been duped. On the other hand, copyright is only concerned with the harm (usually financial) to the author or his estate.
Note: The above information is borrowed from Buranen, L., & Roy, A. M. (1999). Perspectives on Plagiarism and Intellectual Property in a Postmodern World. Albany: State University of New York Press.
Plagiarism is avoided by properly attributing the source. This can be done simply by naming the original source at any point in the presentation or paper. In a paper, an accepted style is often used, such as APA, Chicago, or MLA.
Use of copyrighted material often entails a degree of risk. Broadly conceived, educational/research/nonprofit uses of copyrighted material that adhere to the four factors of fair use may fall outside of infringement. Other uses such as commercial uses will likely involve contacting the copyright holder for permission to use the material in question.