Bike-specific clothing makes for a comfortable ride.
Bike Jerseys
A bike jersey of Lycra® spandex or other form–fitting material reduces drag when you ride. Their fabrics enhance performance by wicking away sweat keeping you cooler and drier. 
Bike Shorts
Cycling specific shorts differ from street shorts primarily by adding stretch for full freedom of movement, and a padded crotch liner(chamois) to reduce friction and wick moisture. If possible, try several different brands on to determine what padding style best fits your anatomy.
Traditional Lycra/Spandex Bike Shorts
This style of short offers full range of motion with minimal weight. There are so many options in this category, that it is impossible to cover it in detail here. Your best bet is to go to the store and try them on, especially because everyone's perceived level of comfort is different. Keep in mind that this style of short is meant to be worn without undergarments. The PWR staff can help you with some advice when purchasing shorts. 
Padded liner: 
A smooth, soft pad of "chamois" (actually made of synthetic material) minimizes friction, wicks moisture, prevents bacterial growth and helps cushion bumps. It's the most complex part of a bike short. There are a multitude of shapes, thicknesses and materials among brands and genders.
Mountain Bike shorts: 
Sometimes called "baggies," these have a loose outer short in addition to the spandex chamois liner. The waist is fastened by a button or hook-and-look patch. Pockets are also common. Choose these by their features and quality of construction, but also make sure the cut of the outer shorts feels comfortable and allows for full leg rotation and flexibility. 

Bib shorts: 
Popular with cycling enthusiasts but a comfortable option for any rider, these don't have an elastic waistband that can restrict breathing. Worn with a jersey, they look like any other bike shorts.

For women, some brands make cycling skorts, where the spandex short is covered by a skirt. 
Bike Tights, Knickers and Leg Warmers:
For cooler temperatures, you may opt for cycling tights, which cover the entire leg, or knickers, which cover the knee and above. Just like shorts, many tights and knickers come with a built-in chamois and should be chosen using the same guidelines for fit and comfort. Tights often include weather-resistant front panels and reflective detailing for dark, winter rides.
For layering purposes, some tights and knickers come without a chamois liner so they will fit over a pair of cycling shorts with no problem. Leg warmers are a handy cycling accessory that can be used on the fly to make a pair of cycling shorts into tights or knickers.
Cycling Jackets:
The top 2 considerations when selecting a cycling jacket: Will it keep me warm? Will it keep me dry? Some styles will do both, but keep the following in mind:
How warm is "warm"? The jacket you select for winter riding in Chicago will probably be different than the one you'd use for winter riding in Phoenix. But don't overdress; you'll warm up from exertion during your ride. Jackets for maximum warmth will protect you against the wind and offer some insulation, mostly in the front and arms.
Mountain bike shoes:
If using clipless pedals, choose shoes that work with your cleats (typically SPD or Look-style) and match your riding style.
Mountain bikers need shoes with durable soles that offer ample tread to grip the trail if needed. It is advised to look at mountain bike shoes which resemble tennis shoes on the bottom. This will help when your rider needs to dismount their bike and climb a rock or other surface with limited traction. Most road bike shoes are smooth on the bottom, which can make it difficult to climb over obstacles if needed. 

Accessory Items:
Caps: These add insulation to your winter rides, while a headband or a thinner skullcap can serve as a sweat barrier and help wick moisture for a cooler head during summer riding.
In summer, gloves with short-cut fingers are the popular choice. Most have a padded leather or synthetic-leather palm and moisture-absorbing terry cloth for dabbing sweat or a runny nose. For cold-weather rides, a pair of wicking, breathable, full-finger bike gloves are a must. Most also offer some protection against the wind. For maximum warmth, consider using a thin liner inside the glove. 
Arm/leg warmers: 
These provide a little extra warmth while taking up minimal space in a shirt pocket or pack. Each is essentially a fleece or wool sleeve that fits over your arms or legs to cover exposed skin. Warmers should be slid on under your jersey and shorts and fit snugly to avoid slippage during a ride. When temperatures rise, they can be easily slipped off without having to unbutton, unzip or change anything. Shop REI's selection of warmers.
PWR staff can assist you with any questions you may have regarding clothing. The choices are plentiful, and it can get confusing.
Thanks to REI for this list of cycling apparel choices.