Achievements are in the "logs" section of the progress tab, and are needed to level up. The Challenges area however, unlocks extra info in the manuals, or badges/some hats at the milestones (5,10,15) in that section (Teaching and citizenship) You can also unlock the newsboy hat by doing all the required actions in a practice match.
The game has an official competitive scene, and also has community-hosted events and tournaments such as the Cogs, the Support your local Aerodrome event, the Sunday rumble, and the saturday box social. Official events have previously included the Faberia Conclusion and the Leviathan Seasonal.
Leaderboards will almost definitely never be implemented, for players or for clans, even though the clan system may be getting an overhaul soon. The bounty board is a competitive 'mini game' in which the bounties try to win as many matches as possible in a set period of time (two weeks) and try to win more than the other bounties. Every player is eligible to become a bounty, by signing up, and out of the pool of applicants, 75 players will be chosen at random to be the bounties. The one with the mist wins at the end of the period gets a badge for their profile page and a prize. The player with the most collected bounty points (earned by winning a match against the enemy team possessing a ship with any bounty on it gets that bounty's bounty points, (bounty points are not claimed for one player, you just cannot claim their points again for the duration of that bounty period.) Also, the player who claims the highest amount of bounty points at the end of the period gets a badge and a prize as well.
Other than the Bounty board, there will almost NEVER be any leaderboards. Also, ranked games will almost never happen, as that would fracture the already small player base.
Summary
Leaderboards — Besides bounty board existing, probably not going to happen.
Achievements — Needed to level up, found in the logs section under the progress tab of the GUI.
Challenges — Complete to unlock extra info in the Journals (Manuals) and some hats and badges, depending on the category.
Competitive matches/tournaments — Official and community-organized tournaments and the competitive scene already exist and are organized through this forum.
Clans — More of an informal system, usually groups of players who use the clan tag functionality to find each other and play together.
Streamed tournaments — Exist, one of the regular streamers's twitch channels is
http://www.twitch.tv/cesports