Common Core seems like two random words that could mean anything. But in the context of education, it is a set of standards in the US that define what a student ought to learn at each grade level from K to 12. In addition, Common Core standards focus on math and English aptitudes only. Also, not all US states are in on these standards. There are only 34 states that fully adopt Common Core in their school systems.
For English aptitude, there are criteria that fall under a number of categories, including phonics, fluency, literature, writing, speaking, listening, vocabulary, and many more. Each student is expected to have a better mastery of each aspect when moving up the grade levels.
As for mathematics, the standards are a bit different. From K through 5, these are assessed: Operations & Algebraic Thinking, Numbers & Operations in Base 10, Measurement & Data, and Geometry. The standards do increase in difficulty each year, though. And for grades 6 through 12, more advanced metrics are used, such as Statistics & Probability and Trigonometric Functions.
Aptitude for the Common Core criteria is evaluated using standardized tests. For each participating state, their methods of evaluation may be different from each other. However, they do work in concert to develop standardized tests. Despite that, states are still flexible in how they want to measure their students' performance.
Common Core has both supporters and detractors. The supporters praise it for providing a set of standards by which they can evaluate how students perform. This used to vary from state to state, making the analysis more confusing on a national level. On the other hand, those against the Common Core say that there is too much emphasis on passing the standardized tests. Even some high achievers don't do well in the tests, and this does not reflect their true performance.