During the conversion process, uranium U3O8 (yellowcake) is converted into UF6 and is shipped to enrichment facilities in its room temperature solid form. UF6 is then heated and converted back into a gas. The two most common enrichment methods are gaseous diffusion and gas centrifugation.

During the process of gaseous diffusion, UF6 gas is forced under pressure through a porous membrane. The difference between the atomic structure of U238 and U235 is the number of neutrons present within the nucleus. U238 has 146 neutrons making it heavier than an atom of U235 with 143 neutrons.
Note: Even though the terminology switches to molecules as opposed to atoms, the concept of atomic weight remains the same. UF6 is comprised of one of uranium's isotopes and F6. Two or more atoms bonded together are called molecules. UF6 molecules that include the radioisotope U238 are heavier than those comprised of U235 due to the difference in the weight of the uranium isotope.
The lighter molecules containing U235 travel through the membrane faster than those molecules with U238. Using pressure to keep the molecules flowing, this process is continued hundreds of times as the gaseous molecules pass through one membrane after another. Eventually a sample with a high enough concentration of U235 (3-5% for LEU) is achieved and the process is stopped.
A method that is fast becoming the most popular enrichment technique, gas centrifugation uses significantly less energy than gaseous diffusion. During gas centifugation UF6 gas is placed into a tall high-speed spinning cylinder. This spinning creates a centrifugal force that pushes the heavier molecules containing U238 towards the wall of the cylinder, while the lighter molecules containing U235 collect near the center. This slightly enriched U235 is collected and fed into the next cylinder and the remaining material in the cylinder is recycled through the process again. This process can occur thousands of times until the uranium is enriched to the desired level.
This is a great video that animates the gas centrifugation enrichment process used by Urenco.

Current Event:
Iran claims its HEU enrichment capabilities are stopping at around 20% and are solely for medical purposes. However, the concern with dual-use enrichment technology is that once a country has begun producing HEU, the step to enriching up to the 90% needed for a weapons-grade product would not be terribly difficult. The negotiations between the P5+1 and Iran led to a 6-month deal reached on January 20, 2014. Iran's enrichment capabilities have been a main factor in these current negotiations. In return for sanctions relief, Iran has agreed not to install any additional gas centrifuges and to stop enriching uranium beyond 5%. They are also required to begin diluting its HEU to LEU as a security precaution as well as confidence building measure for Iran to prove beyond a doubt that its national interests lie solely in peaceful applications of nuclear technology. A report published on February 20, 2014 by the International Atomic Energy Agency said Iran is meeting its commitments under the deal by down-blending its HEU, not installing additional centrifuges, and only enriching to a low level. The P5+1 negotiations, as bumpy as they have been, are a hopeful sign of continued cooperation and an end of current hostilities. However, it will be a deal breaker if either Iran or the member states of the P5+1 do not continue to uphold the bargain.
Sources:
Federation of American Scientists: Centrifuges
Federation of American Scientists: Uranium Production
International Atomic Energy Agency: Nuclear Fuel Cycle
International Atomic Energy Agency: Iran update
Joint Plan of Action
New York Times
Nuclear Chemistry
Nuclear Regulatory Commission
Presentation by Lisa Loden of Oak Ridge National Laboratory