There are multiple missions that you can start from the Command Center, here's a breakdown of all the Fleet Missions available in CosmoCrafter, and how they work.

Attack

You can use your spacecrafts or missile defenses to attack another settlement.
If the settlement is occupied by another player with spacecrafts or defenses, then a battle starts:

First round, you attack and the settler defends on the following order:
1. Orbitals (these defense types have priority when defending the settlement);
2. The Settlers Spacecrafts and/or Standby fleet from another player;
3. Drones, follow;
4. Finally, the last line of defense for a settlement are the Turrets.
The rounds progress with attacks from part to part until there are no more attacker's spacecrafts or settler's defenses/spacecrafts;

If you win the battle against the settler (or if there was no settler in the first place in the settlement), then you can stock up your spacecrafts cargo with resources from the settlement, otherwise, this was the end of your journey to your miscalculated attack.

Additionally, if you're attacking another player, this player will only have visibility on your attack once the spacecrafts are within the capacity of their radar, when this is the case, the player might choose to use interceptors to intercept your spacecrafts.

Colonize

You need to use your Colony Ships to do the trick of colonising an empty settlement, when colonising you can send resources and other spacecrafts to be transferred to your new settlement.
By default the system won't allow you to colonise a settlement that is occupied by another player, but if it happens that your fleet was in transit while the settlement got colonised by another player, then, when your fleet arrives its destination won't do anything and it will return back to your planet.

Coordinate Attack

Same as attacking, but you can do it with your friends. The system will automatically balance the velocity of the spacecrafts from each origin to reach their target at the same time. A combined battle report will be generated to all the users that participated on the attack.

Destroy

The most powerful mission in CosmoCrafter. Destroy can mean two things:

Destroying a settlement: basically entails reducing the level of the buildings of another player's settlement and generating debris on the planets' orbit.
Destroying a planet: quite a challenging task to achieve, you'll need hundreds of thousands of attack power to achieve the result of destroying a planet that basically entails literally splitting the planet into pieces (smaller planets), generating large amounts of debris that will get flying through space and reducing the level of buildings of all the players settling on the planet.
You might be stopped by the defenses of the player or of all the players in the planet.
To do this mission you need to mandatorily send Titan spacecrafts (and boy: you do need quite many to make a meaningful impact and not a fool of yourself).

Extract Resources

Straightforward: choose an empty position on the planet's map and go extract additional resources with your mining vessels. If you try and extract on a settlement occupied by another player, you'll be received with their defenses and might loose your spacecrafts. You can only use Mining Vessels, Scouts and Freighters on this mission type.

Spy

Curious to know what your neighbours have on their arsenals? Send Scouts or Stealth Ships to spy on other player's settlements. Discover their building levels, their resources, spacecrafts and defenses: all you need to know to make a better assessment before attacking. The probability of your spacecrafts being caught spying depends on your Command Center level and the level of your Empire Expansion research.

Standby

Wanna help a friend? Send spacecrafts to standby by another player's settlement and help them defend from attacks. You can standby for a specific period of time or indefinitely (until you issue the order for your spacecrafts to come back).

Transfer

You can transfer spacecrafts between settlements. You can also issue a transfer to another player.

Transport

You can send resources between settlements. You can also send resources to another player's settlement.