How might Christians apply the beatitudes to everyday life?

For being such short passages, the beatitudes are certainly full to the brim with thoughtful material. As far as daily application, they serve several purposes. First, some of them are Excellent reminders of things that one should be ready to treat others with at any moment, each day. The clearest examples of these would be being merciful to others, being "meek" which the American Standard Bible translates as "gentle", and being peacemakers.

Second, they're reminders from God that He has plans outside the temporary world and that He provides for those that are need. It also reminds one that God values the kinds of people that the world often disregards (those who mourn, who are poor in spirit, those who want to be pure.)

As a few final notes, I once had a wise pastor point out something interesting about the beatitudes. "It's easy to see these and think 'wow, I really want those blessings, maybe it would be good for me to be poor in spirit so that I would receive the kingdom of heaven'," he said "But no, being poor in spirit is a difficult and bad place to be! We shouldn't want to 'get poor' to get a blessing, but God is reminding us that everyone who is suffering and is weak is cared for."

The last day-to-day application of these teachings I will mention has to do with being persecuted for righteousness and with being insulted and hated by those who hate God. Jesus ends this teaching by reminding the listener that heavenly rewards are great, and that the prophets long ago were treated in the same way. One may take this knowledge in their daily life and remember that the world will at times persecute believers, but God has blessings taken care of for it, and therefore, do not become sad, don't be unmerciful in response, and do not forget to make peace, because God will cover the damage with an overflow of blessings.