Loyalty Points ("LPs") are awarded by Agents as part of the reward for a successful mission carried out on their behalf. These points accumulate and may be 'cashed in' for favours from that Agent.
These favours take the form of offers of ships and equipment, and all cost a certain amount of Loyalty Points, and sometimes require other goods and minerals.
Declining an offer from your agent doesn't cause loss of standing.
Loyalty Points are not transferable, and can only be used to accept Agent Offers from the agent who granted them in the first place. The amount of Loyalty Points offered in the mission details window can be used as a fair gauge of the difficulty of the mission.