The sequential Phragmen method is a multi-winner election method introduced by Edvard Phragmen in the 1890s.
The quote below taken from the reference Phragmen paper sums up the purpose of the sequential Phragmen method:
The problem that Phragmen’s methods try to solve is that of electing a set of a given numbers of persons from a larger set of candidates. Phragmen discussed this in the context of a parliamentary election in a multi-member constituency; the same problem can, of course, also occur in local elections, but also in many other situations such as electing a board or a committee in an organization.
The sequential Phragmen method is used in the Nominated Proof-of-Stake scheme to elect validators based on their own self-stake and the stake that is voted to them from nominators. It also tries to equalize the weights between the validators after each election round. Since validators are paid equally in Polkadot, it is important that the stake behind each validator is spread out. The equalization method is ran twice for every validator election. The first iteration will do a rough equalization among all validator candidates in order to determine the subset that will become the next active validators. The second iteration runs only among the elected candidates to equalize the stake between the ones which are elected.
The Phragmen method is also used in the council election mechanism. When you vote for council members, you can select up to 16 different candidates, and then place a reserved bond which is the weight of your vote. Phragmen will run once on every election to determine the top candidates to assume council positions and then again amongst the top candidates to equalize the weight of the votes behind them as much as possible.
Phragmen is something that will run in the background and requires no extra effort from you. However, it is good to understand how it works since it means that not all the stake you've been nominated will end up on your validator after an election. Nominators are likely to nominate a few different validators that they trust will do a good job operating their nodes.
You can use the offline-phragmen script for predicting the outcome of a validator election ahead of a new era beginning.
W3F Research Page on Sequential Phragmen Method - The formal adaptation of the Phragmen method as applied to Polkadot validators.
Python Reference Implementations - Implementations of Simple and Complicated Phragmen methods.
Substrate Implementation - Rust implementation used in the Substrate Runtime Module Library.
Phragmen's and Thiele's Election Methods - 95-page paper explaining Phragmen's election methods in detail.
Offline Phragmen - Script to generate the Phragmen validator election outcome before the start of an era.