Also, I would like to add a little something for same-sex US/EU partners who registered or will be registering a civil partnership in the UK. All of the above information applies, except for when it is time to apply for Further Leave to Remain (FLR). If your EU partner has been present and settled in the UK for 5 years and has been granted Indefinite Leave to Remain (ILR/Permanent Residence) by the HO, then the US partner is eligible to apply for Further Leave to Remain using Form FLR (M). This will give the US partner a 2 year visa and they will be eligible for ILR after the 2 years is up and eligible for British Citizenship after 1 year on ILR.
If your EU partner has not been present and settled in the UK for 5 years, or has never officially applied for ILR/Permanent Residence through the the HO, then the US partner is eligible to apply for a residence card using Form EEA2. The residence card is valid for 5 years and allows you to live and work in the UK. This application can only be applied for by post and there is no processing fee. Once the application is received by the HO, you should receive a letter confirming this within a couple of weeks and telling you that it can take up to 6 months to process. While you are waiting for your application to be processed, the letter should inform you that you are eligible to seek employment from the date of the letter (so make certified copies of your passport before you send it in and use that along with the letter as proof of employment eligibility).
After the US partner's 5 years is up, they will be eligible to apply for permanent residency (ILR) by using Form EEA4. They will be eligible for British Citizenship after 1 year on ILR.