I've run a lot of redundancy programmes, it's all slightly different but in the main...
Behind the scenes they will decide what the new structure should look like - so if they have to reduce costs by 20% for example their best idea of how they can do that and still continue working.
This will usually mean reducing or removing any activities/roles that are 'nice to have' rather than absolutely critical for that organisation. It may be reducing team sizes and/or asking managers to manage bigger teams.
Then, when this has all been done they announce the changes that are proposed.
Depending on how many people are being made redundant they may or may not need to have a period of consultation with staff at risk. Under 12 people impacted = no consultation. After that it's a 30 day period unless more than 100 people impacted as then 45 days.
If all of a certain role are being removed (let's say every single person doing the ABC Role) then at the end of the consultation period those roles are made redundant. You will be told of any suitable alternative roles in the organisation that you can apply for but otherwise (or if not successful in applying for them) you will be given notice.
If it's not the case that ALL of a certain role are being removed but they just reduce the number of those roles (so a team of 10 ABC Roles now needs to be a team of 5) then they can do one of the following.
Paper process; They come up with a scoring matrix and assess all ten people against it. Those 5 with the best scores keep their jobs. They can decide on what is used for scoring but it has to be objective so is usually some combination of previous performance scores, previous sickness/absence records and any disciplinary issues.
If there is a tie they will usually then go to interviews between the people tied.
OR
Interviews; They decide not to do a paper process and all 10 people are interviewed for their roles. 5 are successful, the other 5 are redundant.
Occasionally they may ask for 'volunteers' before the paper/interview process. This doesn't happen so often these days but I believe still happens at universities sometimes.