We highly suggest that you familiarize yourself with the entire process and documents needed before you submit your online application because once you do, you start a clock and only have a limited amount of time to finish the subsequent steps.  If you miss the deadline, you'll have to apply from the beginning again including paying a second time.  

Thanks to Aquila for this comprehensive list.

Only submit the online application once you are totally ready to go, know that you meet the requirements, and have everything required and ready to send.

The application is made online:  
https://www.visa4uk.fco.gov.uk/

The general process of applying is as follows:

1. Gather all your supporting documents .  Do not submit the application until you have all your supporting documents in hand.  When you submit your application, you will schedule and attend a biometrics appointment.  Once you have attended your biometrics appointment, you only have 14 calender days to mail your visa application package to the UK (or 30 days from when you submitted the online application - whichever is sooner).  If you miss the deadline, you will need to submit a new application, including paying the fee again.  All your documents must be originals or certified copies.  If you want an original document returned, include a regular photocopy of that document with the original.

2. Pay the NHS levy,  a healthcare surcharge, for yourself and dependants before submitting your visa application. 

https://www.gov.uk/healthcare-immigration-application/overview

3. Complete the application online .  At the end, you will pay the fee and can book a biometrics appointment (where you get your fingerprints and picture taken).  The biometrics appointment takes place at a USCIS center, and you can choose any one.  (the system will try and offer centers closest to you, but you can choose other ones if it suits you better - ex. a center closer to work)

4. Once you have completed biometrics , mail in your application package including your supporting documents to the UK.  (disregard information to send the application package to the UK consulate in the US; this is outdated).  If you wish to have your application processed as priority, you purchase the service on the day you mail your package.  You may receive an email when your application is received (the system is a bit buggy; not everyone does so don't panic if you don't get an email).  

5. Wait for a decision (probably the hardest part!).  Processing times can be highly variable depending on the volume of applications at the time.  The wait can be a few days, a few weeks, or possibly a few months.  Priority processing will ensure your application is reviewed quicker; however, there is no guarantee of how soon it will be done.  If there are a high volume of applications at the time, even with priority processing, the wait can still be a few weeks or more.  You can check out some of threads where people have been sharing their processing times to get an estimate of how long it's been taking recently.

6. When a decision is made , you will get notified by email and your documents will be returned to you.  (the email won't say if the decision is yes or no, just that a decision has been reached... you'll get the actual decision in the package that's mailed back to you).  If approved, your passport will have a visa sticker attached inside.  Check the visa sticker carefully and ensure there are no errors.  If there are any errors, you may need to send your passport back so a corrected visa can be issued.  If not approved, you will receive a letter detailing the reasons why the visa was not approved.

7. Once you have your approved and correct visa , you can book your flight to the UK.  Due to variable processing times and the potential for needing a visa error corrected, it is highly recommended to not book any flights until the visa is issued and inspected for accuracy.

Check out the following links for more info:

General Info:  
https://www.gov.uk/join-family-in-uk

Financial Requirement:
Annex_FM_1_7_Financial_Requirement.pdf

UK Visa Financial Requirement Calculator

UK Family Migration Financial Evidence Checklist

Example Application Form to use as a guide (do not use to apply; you must apply online):
VAF4A.pdf

Appendix 2:
VAF4A-Appendix2.pdf

Appendix  FM-SE specified evidence
20150108_Immigration_Rules_-_Appendix_FM_SE_final.pdf

Supporting Documents Government List:
sup-docs-settlement.pdf

How Long Will it Take to Approve or Process my UK Visa Application?

Processing time is variable; we've seen people waiting a few days and at other times, people waiting a few months.  You can check out visa forum threads detailing people's recent experiences to get a feel of timelines.  If you want your visa application to be processed faster, you can purchase priority service - it's optional and not buying it will not have any effect on your visa decision.  You should purchase the service on the day you send your application to Sheffield.

By using priority processing, you are having your application put ahead in the queue, so it will be processed ahead of applications without priority processing.  The processing times can vary depending on how many applications are pending decisions, so there is no guarantee of when a decision will be reached.  Historically, priority decisions have taken anywhere from several days to several weeks.  Sometimes there have been significant differences in processing times between priority and non-priority applications, and sometimes the processing times have been very similar between the two.  

Anyone can purchase the service; however, you should expect longer processing times (even with priority) if any of the following apply to your application: 
 
- previous refused UK visa, 
- previous refused leave to enter the UK, 
- previously been removed or deported from the UK, 
- overstayed a period of leave in the UK, 
- been refused leave to remain in the UK, and/or 
- have any unspent criminal convictions in any country.  

If one or more of these conditions apply to you, you are advised to not purchase priority processing, or to expect a decision to take longer than usual (even with the priority processing service).

Priority Processing:  
http://www.vfsglobal.co.uk/usa/user_pay_services.html

You can ask on the application what date you want the visa to be active, but usually, the visa is active from the day it's approved.  You can apply for the visa up to a maximum of 3 months before your intended travel date.

Thanks to Aquila for this comprehensive list.