Colombian with a UK residence travelling to Colombia through Miami

As I think you know, and as detailed in other questions, sterile international transit is not possible via the United States. Even if you are getting on a plane out of the country as soon as you arrive, you must still go through immigration and customs screening to enter the U.S. The U.S. Visa Waiver Program is based on citizenship, and Colombia is not a VWP country, so you correctly surmise that you would not apply for an ESTA.

Preamble aside, the general U.S. transit visa is a C1 visa. You can also use a B1 or B2 visa for transit, if you have one. While you are supposed to consult the U.S. Embassy in London website for exact procedures for UK residents, their website's links for applications and required documents appear to be broken. The basic procedures for all non-immigrant visas are similar, however:

1. Complete Form DS-160 online.
2. Have a digitized photograph that conforms to the photo requirements and upload it as part of your application.
3. Print out the confirmation page (with the barcode) to bring to your interview.
4. Pay the application fee, currently 160 USD.
5. Schedule an in-person interview at the embassy in London or the consulate in Belfast (the consulates in Edinburgh and Cardiff do not handle visas).
6. Assemble the required documents, which will include your passport (which must be valid for at least six months after your stop in Miami, and have at least one blank page where the visa would be affixed), the DS-160 confirmation page from #3, the payment receipt from #4, and two photographs (if you were unable to upload a suitable photo when completing Form DS-160). Additionally, you will need supporting documents which demonstrate your travel intentions, at the bare minimum, printouts of your travel itinerary and airline ticket receipt.
7. Attend your interview, bringing the required documents. Your fingerprints will be taken, and you will leave your passport with them until processed.

Processing time will vary; at this writing, the estimated wait is 8 calendar days at London, 32 at Belfast.

If you're asking yourself, "isn't this insanity— all this just to be able to change planes?" then without editorializing too much, I would say the answer is yes. The procedure is essentially the same as applying for a six-month B2 visitor visa. But international-to-international transit passengers are, now and historically, a very low priority for U.S. transportation planners, and an even lower priority for U.S. politicians, so this is the situation for the foreseeable future. I invite you to come spend time in the U.S. as a visitor, but for international-to-international connections, look into alternatives. Canada also does not have sterile transit, for example, but at least their transit visa is less laborious to obtain.