The public tender
The need for a go anywhere reconnaissance vehicle was being felt by the Army since World War I and this promoted the appearance of several proposals over the years. After several tentative versions the specifications laid down by the Quartermaster Corps, on July 7, 1940, called for: 
1. A driving front axle with 2-speed transfer case including provisions for disengaging the front axle drive. 
2. A body of rectangular design with a folding windshield and 3 bucket seats. 
3. Increased engine power (presumably in respect to the Belly-flopper prototype). 
4. Means for towing. 
5. 30-caliber machine gun mount. 
6. Blackout lighting. 
7. Oil-bath air cleaner. 
8. Hydraulic brakes. 
9. Full floating axles. 
10. Wheelbase of 80". 
11. Maximum height of 40". 
12. Maximum weight of 1275 lbs. 
13. Approach and departure angles of 45 and 40 degrees, respectively. 
14. Must reach 50 mph on hard surface. 
15. Special bracing for a pintle hook setup. 
16. No aluminum to be used for cylinder head. 
17. At least 4 cylinders. 
18. 8 of the 70 vehicles to made had to be four-wheel-steering. 

Ford, American Bantam and Willys-Overland answered the call.
Banton built the first Jeep in 1940. It was the Bantam Pilot (BRC-60).
Seventy of these units were produced.
Ford produced two units called the PYGMY.
In 1941 they produced 3700 units called the Ford GP
Willys produced 1553 Model MA’s
From there the rest is history.

The name Jeep also has an interesting story to it.
The first coinage of the word "JEEP", as applied to a motor vehicle, occurred during WWI. According to Major E.P. Hogan, who wrote a history of the development of the Jeep for the Army's Quartermaster review in 1941. The word "Jeep", he wrote. "is an old Army grease monkey term that dates back to WWI and was used by shop mechanics in referring to any new motor vehicle received for a test." Jeep was still used in Army motor pools well into the 1930's as general purpose or "GP" for short. Though in a 1941 Army manual it refers "GP" as Government 80" wheel base car (Jeep). 
Before then in 1936, the term 'Jeep' was it introduced to the world by a Popeye comic strip character, "Eugene the Jeep". The only words Eugene could say were "Jeep, Jeep", and "GP" then became "Jeep". Then the "Willys Quarter-Ton Jeep was 1st used by the U.S. military in WWII. When the WWII Jeep was issued during wartime, the term "Jeep" was here to stay.
 

A further note to the history of the Jeep was provided by Rodney Hawk of  Butler, Pa.
" The American Bantam Car Works of Butler, Pa answered the call of the army and designed and successfully tested and demonstrated the jeep for the Army.  Willy's Overland and Fords entrys did not meet the Army's specs.   American Bantam had a very limited production capacity so the army took Bantams entry and gave it to Willy's and Ford to produce.  Unfortunately there was no recognition or compensation for American Bantam and the company shortly went bankrupt.'
To read more about the American Bantam Car Works click on the link below.
Rodney also provided me with two web sites that tell the real story about the Jeeps beginnings

Four of us were riding out to relieve the guard post at an airport some miles out of town, when our jeep slid off the road into the ditch.  Jeeps are famous for being able to claw their way out of almost anything, but not this time. 
With lots of four-wheel spinning, and three husky men tugging, the jeep was in for the duration.

Along came a farmer with a horse and a sleigh.  The horse was a magnificent animal, very broad across the hind-quarters.  By means of sign language, we were able to get the farmer to hitch his horse to the jeep.

Now comes the interesting part.  The horse was standing on smooth, hard, glare ice.
To gain traction, the arctic horse stood up on its very tippy-toes, the front points of its hooves, and promptly walked off with the jeep as if it were nothing.

The moral of this story is...

"Hey mister, Get a horse."