Tier 1 link building refers to establishing links between various websites and the website you are trying to optimize for search engines. This is the most common type of link building.
Example: Website A -> SEO Target Website
In some cases, especially when building links that might not be considered high quality, it makes sense to build the links to an intermediary website before establishing a link from that website and on to the website you are trying to optimize for search engines.
Example: Website A -> Website B -> SEO Target Website
Tier 3 link building is the same concept, just with a third layer of intermediary websites to further negate the risk of passing on 'bad link juice' to the website you are trying to optimize for search engines.
Example: Website A -> Website B -> Website C -> SEO Target Website