There are two physical methods of building a pure fission nuclear weapon:
    Gun Assembly, where one mass of fissile uranium is fired through a tube at
    another mass so that together they create a chain-reacting critical mass.
    The atoms must be "smacked" together quickly so the reaction doesn't just taper on.
    Implosion style, where concentric fissile material, surrounded by high explosives is
    explosively compressed into a chain-reacting critical mass.

Little Boy was a Gun Assembly, while Fat Man was an Implosion device.   The fuel for fission weapons may be either Uranium or Plutonium, the Implosion type can use either material, but the Gun Assembly type only uses Uranium due to decay factors of Plutonium.  
The crux of this is that several sub-critical masses of nuclear fuel must be brought together in a very short space of time, so that a self sustaining nuclear reaction takes place.   Another factor, the two masses must be not just close to each other but must be impacted together similar to a compression weld so the atoms are close enough that the probability of neutron collision is very high.   As the reaction builds, material is converted to engery, which must escape, like the energy created at the sun's core, only in these small reactions the reacting material is forced apart until the reaction stops due to no proximity (percentage of particle collisions reduced to near zero).

Fusion versus Fission:
No pure fusion weapon has been invented to my knowledge.   But a fusion process is used to enhance the reaction taking place in a fission explosion.   Tritium or Deuterium is added to the core of an implosion weapon so that it is compressed and fuses by the fission reaction.   This fusion causes the emission of more neutrons by the fusion fuel which increases the number of neutrons available to fission the fission fuel.   This, in turn, causes a more complete fission of the available material, a larger explosion if you will.   The fusion explosion adds very little explosion wise, but by consuming a larger part of the fission fuel, greatly inhances the explosion

It occurred to me to use more than two pieces of mass, however this really complicates the mechanical structure and detonation timing of a Gun Assembly weapon, where do you put the pieces before the detonation, remember this is a three dimensional problem?   Could you build a 3 or more barreled cannon that all fired at a common point with precise timing so that all the pieces of fuel reached the junction at precisely the same time?   At least the multiple pieces would not be in line of sight of each other, but the hollow tubes (barrels) would allow a lot of interaction before detonation.   A similar approach in an implosion weapon would be multiple concentric hollow spheres.   I always wondered why the Gun Assembly has the Plutonium deterioration problem and the hollow spheres don't?

A 60 foot tall Redstone nuclear missile, outside the building.   I believe White Sands Museum also has a Redstone.

A pair of Navy Terrier missiles.   The Terrier was the Navy's first operational surface to air missile.

A model of B-29 Enola Gay - dropped Little Boy on Hiroshima.

Model of B-29 BocksCar - dropped Fat Man on Nagasaki

Norden Bomb Sight.

Norden Bomb Sight.

A pair of Mark-28 thermo nuclear (hydrogen) bombs lost in the Mediterranean sea.   These could be carried by fighter aircraft, houndog and mace missiles.

The Mace missile can carry a Mark-28 Thermo Nuclear (hydrogen) warhead.

Matador surface to surface missile, deployed by the U.S. Air Force, and capable os conventionl or nuclear warhead (W-5).   Len: 39ft 6in, Span: 28ft 7in. Dia: 4ft 6in. Wt. 12,000, Rng: 700mi, Spd: 650 mph.   It was phased out in favor of the "Mace".

A Walleye glide bomb, sometime called "Fat Albert", can be carried by jet fighters.

The large, reddish brown missile up on the rack is a Trident-2, submarine launched.

A MK-77 incindiary bomb.

A Minuteman Nuclear Missile, launched from hardened underground silos.

A Nike Herculese ground to air missile once used to protect U.S. cities.

280 mm Atomic Cannon, largest nuclear capable, mobile artillery piece manufactured in US.   On May 25, 1953 a 280 mm atomic cannon fired an atomic projectile 7 miles at the Nevada Test Site.   Twenty of these were made.   Len: 84 ft, Wt: 86 tons, Range: 15-19 mi., warhead: W-9, w-19.   Link to youtube video of cannon being fired   Link to slightly smaller pic.

Genie an air to air nuclear missile to protect the US from large numbers of enemy bombers.

A Little Boy casing, note the radar detonator antenna (shiny metal loop sticking out from white block near the front).   Little Boy was called a "gun assembly" device.

A Fat Man casing, again note the radar detonator antennas, shiny metal loop protruding near the middle.   Fat Man was an "implosion" device.

Honest John Missile.   From Wikipedia: The MGR-1 Honest John rocket was the first nuclear-capable surface-to-surface missile in the US arsenal.[notes 1]   Designated Artillery Rocket XM31, the first such rocket was tested 29 June 1951 and the first production rounds were delivered in January 1953.   The designator was changed to M31 in September 1953.   The first Army units received their rockets by year's end and Honest John battalions were deployed in Europe in the Spring of 1954.   Alternatively, the rocket was designed to be capable of carrying an ordinary high-explosive warhead weighing 1500 pounds, even though that was not the primary purpose for which it was originally envisioned.

Lower conning tower from SSN James K. Polk, Nuclear Submarine, being resotred by the museum.   You can also see the diving planes to the far left.

SSN James K Polk at sea