In Anthem, when your health drops to zero, you need to wait for your squad members to revive you. This is not always possible or makes your teammates vulnerable, e.g. during a boss fight. In such cases, after several seconds you can use the Respawn ability by holding the Y key. This will bring you back to life at the nearest checkpoint or Strider (in the Freeplay mode).
Respawning is not always possible! When completing story missions, contracts, engaging difficult battles (especially the ones with bosses), you will be informed about the respawn restriction. In such situation, should you fall, only a squad member can help you up (you cannot even disconnect from the session as long as you're down). Should an entire squad fall, all of you will resume at the nearest checkpoint.
Strongholds form an exception from the above rules - the entire area of such mission is a respawn restricted area. It is worth to actively help the knocked-down players while completing a Stronghold, especially since, due to their high level of difficulty, the execution of mission goals may be much more difficult in an incomplete line-up.