I hope to continue to add to this list as I come across better/worse options
So you have your circuit designed and ready to go, but now where to get your components from?
First lets take a look at what you're getting. My advice, if you're just getting a few basic parts; resistors, capacitors, switches, LED's, etc, check out Radioshack first. There is usually one close by where ever you are and they care most basic components. This saves you on shipping. Otherwise if you're going to make an online purchase, here's my recommendation.
Newark - Element 14 (Link Here)
Newark is my favorite distributor for electronics. They have a huge inventory of parts on a wide range of uses. Almost always have what I'm looking for. They are usually very well up-to-date with data sheets and product information. They have online chat or phone support for their website or technical information. Their website is easy to navigate to find what you are looking for and add requirement filters to your searches. They also allow you to look up similar parts based on specific requirements. You can also import a Bill Of Materials or CSV file to quickly fill your shopping cart. They allow for saving shopping carts for if you ever wanted to re-purchase a design. Newark also usually has a few sales posted on their homepage that you can take advantage of. I really have never had an issue with Newark and I highly recommend them.
Digikey (Link Here)
I used Digikey for a while before I started using Newark. While their website may not be as attractive as Newark's they have a quality inventory of parts. Some may find that their website is easier to navigate. I found it difficult to find certain specs on their product pages. They also have live online chatting for assistance. Digikey's shopping cart saving options is probably the best handled. They provide ID numbers for purchases/shopping carts, along with salesman ID's, so you can literally make the exact same purchase. I find the better detail makes for better record keeping. All in All Digikey is another great site for purchasing components.
Mouser (Link Here)
Mouser is another option online. I'm not a huge fan of their website, but some people do prefer it. I don't think they have a selection as large as Digikey and Newark do and I've had some difficulty finding some components on their site before. They do have a new "Intelligent Bill Of Materials Import Tool" which is worth trying out. It makes organizing your order pretty easy. So if you aren't looking for really specific components, and you want to try their ordering tools out, It's work it to check them out and give it a shot. Not my personally preference because I like rooting around and picking really specific parts, but some people prefer different things.
Allied Electronics (Link Here)
Allied Electronics has a flashy site that is easy to navigate, and they also have a decent inventory. They seem to have a wide range of components, but to navigate to specific ones may take longer and more searching/filtering than Newark's site. I've had issues with not being able to find specific parts on their site before, so I haven't used them all that much and I cannot comment on their entire ordering process, but from what I have seen I don't think they have any kind of cart saving or BOM uploading options.
SparkFunhttp://www.sparkfun.com/ (Link Here)
Sparkfun is a great website for simply DIY and more advanced DIY projects. They provide great easy to use breakout boards that allow you to connect to all kinds of sensors and components on either PCB designs or breadboards. They make their products super simple to use. They provide and Eagle library for all of their products and they usually have an example with each product to give you a little shove with getting it to work. I like using Sparkfun for little projects or even projects that you don't need to know everything about you just need to get it running. They have a wide variety of compatibility for different micro-controllers and usually have a lot of starter kits to make learning them simple.
So here are some popular sites that you can use to purchase any project components you may need. Some site are better than other and all sites are always up for personal criticism. Try them out and choose whichever one works best for you. They all have their pros and cons. I personally choose Newark more often than the other, but you are obviously welcome to use whichever you like best.