What are the achievements of Caliph Umar?

Umar's greatest achievement was undoubtedly the expansion of the Islamic empire. Under his leadership, Islam spread to new territories in what is now the Middle East, and it's a testament to his leadership that these territories remain part of the Islamic world to this day.
This was partly because Umar ordered his invading armies not to force conversion on non-Muslims. He understood that Islam was much more likely to take root if people actively chose to embrace the faith without being coerced. This strategy paid off handsomely, with vast swathes of what had previously been pagan lands converting to Islam.
Once he'd conquered new territories, Umar immediately set about establishing institutions of law and government. This greatly contributed to the stability of the rapidly-developing Islamic world and ensured that everyone, Muslim and non-Muslim alike, was given the full protection of the law. Umar also revolutionized the system of land ownership by setting up a Land Revenue Department that ordered the survey and assessment of the new lands that he had conquered. This ensured that the growing additions to the Islamic empire could be managed and maintained more effectively.