Your character in EVE Online is represented as a clone, an immortal whose conscience is transferred onto another clone upon death or when clone-jumping to an alt clone placed in another station. Clones are split into two states:
The Alpha Clone state is the base state for all characters, providing open access to the EVE Online service and enabling the free exploration of the New Eden Universe. It is available to any character in New Eden at any time. Alpha Clones will be able to train a specific set of skills including tech one Frigates, Destroyers, Cruisers, Battlecruisers and Battleships for all factions, along with essential weapons and modules.
Alpha Clones train skills at 1x speed and have a limited set of skills (all Omega skills & skill levels will be marked with a yellow highlight ingame). As an Alpha clone - you can freely train any of the Alpha skills up to 5.000.000 skill points after which the active training stops, however - by using regular or daily Alpha skill injectors, an Alpha clone can accumulate up to 20.000.000 skillpoints.
The Omega Clone state provides broader access to the features of the EVE Online service and will become active as soon as any game time is added to the account in any capacity. Characters on such an account will have unlimited skill access with increased skill training speed, as well as access to all other game features unlocked by these skills.
Upgrading an account from Alpha to Omega is effective immediately and will not require to log out and back in.
If the Omega game time on an account lapses and the account reverts back to an Alpha state, any trained skills and skill levels outside the Alpha skill set limits of the character will become inactive on the next log in. They will become available again as soon as the account receives additional game time and changes to the Omega state again.
To unlock the full potential of your clone, be able to actively train above 5.000.000 skill points and use skill points above the 20.000.000 limit - you will need to upgrade to the Omega state by:
Note that there are multiple packs available for EVE Online that can be applied once per account. These DLCs come at a discounted price and contain various extra items like ship SKINs, extra PLEX, multiple month Omega state, multiple character training and skill extractors.
So if you haven't gotten any - these would be a great way to get some extra bang for your buck.