To "measure up" means to be good enough. For example: "As much as he tried, John could never measure up to the high standards set by his older brother." To "measure" usually means to determine the size or capacity of something. For example: "We need to measure the size of the room carefully to see whether the new rug will fit." [Note that measure is also a noun with more than one meaning.]

Measure up = to be acceptable. "I'm so glad that I measured up compared to everybody else" = I'm so glad that I (was as acceptable as) everybody else To measure = to measure. Examples: I measured the width of the door, and the width of the couch, and I knew the couch would fit through the door. Every good carpenter measures twice before cutting anything.