I received my second British passport (ie; simultaneously valid with my other one) through the post recently so I thought I'd share what to expect.

I rang up the passport office and asked what the procedure was, they told me that I could only get one for business purposes and needed a letter from my company explaining why I needed one. I asked if they were sure about this as I'd heard of people getting one for personal visits, he asked his boss who confirmed that I was right.
He then told me to send off a normal passport application filled out as if I was getting my first adult passport and to include a letter stating why I needed a second passport.
I did so, and assumed that I needed to include all the relevant supporting documents (birth certificates etc) though I was later told this was not necessary. (I had put my reason for needing a second passport as "Visiting incompatible countries" as apparently applying for multiple visas simultaneously is no longer an acceptable reason)

Unfortunately it was apparently insufficient and I was called by a rather nice lady at the passport office telling me I'd need to supply a complete itinerary of my travels and proof of travel.
I pointed out that the only proof of travel I could possibly supply (as my trip is from the UK to India) would be a ferry booking, and I was told this was sufficient!

I wrote up a letter explaining my plans (I had no idea when I'd be going through what countries so I made up some dates that fitted with the sizes of the countries and my ferry booking) which you can find here.
Application letter #2

After sending that letter off I was pleased to receive my second passport shortly afterwards.

So to get your second passport send off the following

1) Proof of travel (ferry booking ticket or something)
2) A standard "new passport" application form filled out as if it were your first passport (I still chose to include my current passport number in the "Other passports held" section of the form (no supporting documents necessary)
3) A letter stating your itinerary down to the dates of travel through each individual country (obviously including the incompatible countries in this list!)
4) Postal order (not applicable if you're using the post-office passport checking service which allows you to pay then and there by debit card)

Then you should get your second fully legal passport with no issues at all!