To receive the old age or disability pension, apply for it via the Social Insurance Board. A person is eligible for an Estonian retirement pension when they have proof of at least one year of pensionable service in the EU member states, Norway, Iceland, Liechtenstein or Switzerland.
Furthermore, Estonia has separate international contracts with Latvia, Lithuania, Finland, Ukraine, Canada, and Russia.
Social care: summed up
After you have moved to Estonia and acquired the right of residence or a residence permit, you will become eligible for Estonian health insurance with all the accompanying benefits. You can also opt for private health insurance should you want so.
All people working in Estonia – self-employed or otherwise – can take advantage of contribution-based benefits (check the Social Insurance Board for most of them). Upon retirement, you will then also receive a retirement pension.
Estonian government agencies have comprehensive websites in English (and usually Russian) that are easy to navigate. For the best overview of all your requirements and rights, have a look at the citizen portal eesti.ee or the website of the European Commission.