Cargo airlines are known to regularly use planes designed primarily for passenger travel.
Why don't they almost exclusively use aircraft designed specifically for cargo?
Price, commonality, and size choice.
A very large proportion of an aircraft's cost is in design and certification. It's followed by the engines, the avionics, and the fuselage. Not much cost can be removed from an aircraft by removing the passenger-specific elements of design. That is done in freighter versions.
Freighter models like the 747-8F already realize most of the possible savings. Second-hand airliners are cheaper still, and they're not hard to convert. Rip out the interior, install cargo securing systems, and fly.
Commonality with airliners means airliner-derived freighters are easier to service and repair anywhere one needs to. And, as Dohn Joe rightly adds, get pilots for; cargo airlines are both a training ground for passenger pilots and a career path of its own.
Finally, there are more airliner models than freight-only aircraft models, so there's more choice of the right payload and range capability.
This reuse of passenger airliners for freight is a great example of efficiency. Passenger airlines tend to have very high utilization, putting lots of cycles on their airframes and lots of fuel through them, so they care for better fuel efficiency, availability, and passenger experience (or at least their expectations when booking, considering the 9-abreast 787) delivered by newer airframes.
Cargo airlines run looser schedules, because their clients don't care much for their flights leaving on time (air freight is usually offered in 2-5 days), which shifts the optimization focus from fuel efficiency towards capital costs. They are able to use up the rest of the airframe's life in a less-critical job, where the oversight is a bit more forgiving and delays due to lower availability are more acceptable.
The An-225 is special in that it occupies a class above every other freighter in capability. It does the jobs nothing else can, but, until recently, the only airframe built has been able to serve the whole world.
Pure freight operations tend to fly significantly fewer cycles than passenger or combi aircraft. And you don't care too much how ratty looking it is as long as it's reasonably reliable. This makes older used aircraft a lot more attractive from a business case perspective. You have airlines that buy combination pax/freighter aircraft new, "combies" but these are still primarily pax aircraft that are able to carry paying freight rather than go partly empty on a route with a low load factor at certain times of the day.
The best source of used aircraft is pax aircraft being retired by airlines, before they're fully worn out, when they upgrade their fleets. Many normal passenger airliners have freighter conversion kits available by Supplemental Type Certificate for large cargo doors. So a old MD-11 that is at 70% of its airframe life and can be had really cheap, still has quite a lot of life left in it if it carries on as a pure freighter at maybe only 4 or 500 cycles per year. With the modest cost to convert to a freighter (cargo door and interior) it makes for a really attractive business case.
If you paid the big bucks for a new pure freighter (as opposed to a combi, which is still primarily a passenger a/c), the capital costs would kill you; you have to fly the heck out of it to make enough cash flow to justify the capital costs and freight operations just don't have the frequency.
There is no such thing as aircraft designed specifically for cargo.
The types like C-5 Galaxy, C-17 Globemaster, An-124 Ruslan, C-130 Hercules or A400M Atlas are not designed for just any cargo, but for carrying cargo to the front line in a war, with ability to land on a strip prepared in a day by a small team of battle engineers and option to para-drop cargo and troops. That's a lot of very special military requirements a commercial operator flying between properly equipped airports has absolutely no use for, and all of them reduce fuel efficiency and increase maintenance costs. That's why nobody ever ordered a Lockheed L-500, the civilian version of Galaxy.
For operating between reasonably equipped airports, the only difference between cargo and passengers is that passengers want windows, and some equipment in the cabin, and cargo needs bigger door to be efficiently loaded. So there are no specific cargo designs, just aircraft built or converted for cargo, without (most) windows, but with big door for easy loading.
The exceptions are:
Another factor is that airplanes (like almost anything else) become cheaper when you buy them in bulk. That means a massive capital investment up front. FedEx, for instance, was ready to buy 10 A380s at one point, but backed out due to production delays
FedEx Express, the express package delivery unit of U.S. shipping company FedEx, became the first customer to terminate an order for Airbus’ flagship plane when it scrapped an agreement to buy 10 A380-800F aircraft.
Instead, FedEx ordered 15 Boeing 777 Freighter aircraft and took options to purchase 15 more. The previous agreement with Airbus included an option for another 10 A380 planes, a FedEx spokesman said, but those options are now invalid.
FedEx said it expects to take delivery of four Boeing 777s in 2009, eight in 2010 and the remainder in 2011. At a list price of \$232.5 million to \$240 million each, the order will be worth at least \$3.48 billion to Boeing.
It's worth noting that FedEx Express flies their fleet almost constantly, which means they're getting a lot of value out of buying brand new. For most other airlines (like, say, National Airlines, which flies only on-demand air cargo) it makes better economic sense to rehab a passenger aircraft.
These days airlines often transport passengers by day and cargo by night. Seats are mounted on pallets, they can all be removed in about half an hour. Then the cargo goes in. The margins for passengers are shrinking but those for packages go up.
The Type of aircraft is called combi, one Example would be Boeing 747-400M, there are more current types.