In this case it was decided that the terms of segregation could be met without violating the constitution. The part of the law that was satisfied was the 14th Amendment of the US Constitution. This amendment granted full voting rights to former slaves. It was ratified in in 1868. So Plessy somehow didn't violate the intent of the 14th. It did that by finding that the notion of Separate but Equal did not violate the constitution.
Brown v Board of Education May 17 1954 overturned Plessy and went back to the original intent of the 14th. Brown v Board of Education ruled that the state (Kansas) failed to protect its citizens and insure that everyone under the law had an equal right to share in the services (education in this case) offered by the state.
