Figuring out what type of person you're looking to hire is pretty easy. There are a number of different roles people play in the development of a website. You have Designers, Marketers, Front-end Developers, Back-End Developers and Database Developers. The type of person you need will depend on where you are within the development process. I'm going to go over some of the most common spots I find websites struggle in. This includes design, development, and search engine optimization or marketing.
The most common website developed is one that has little to nothing to start with. You may not even have a logo design at this point. If you are here, you'll want to start off with a designer. Have them get you a Logo and set up your color scheme. From there, you'll need to work with a Marketer to get your Search Engine Optimization started. Then, a Front-End Developer can take your design assets and SEO and create your actual website. If your website includes a shopping cart or a database for any reason, you'll then want to hire a Back-End Developer to tie it all together. If your website is large enough, you may even consider hiring a Database Developer to help better organize your information and data.
A number of websites I've found are in this phase. The design is either completely done or partially done. In some instances no current code can be salvaged in a rebuild. In other situations, a good Back-End Developer may be able to get you right on track. This will depend on the developer you hire and his skill set when compared to the project. Years ago, there was no distinction between Front or Back End Developers, we were just called Developers and defined by the languages we used. Many older web developers, those that have been doing development since 2005 or earlier, should have all the development skills you require to complete your project.
Typically this is being done long after a website is built and the owner is looking to get more search traffic. This could call for a complete remodel, but often what a client has will work. If your website is completely dynamic, you may need to hire a Back End Developer to add in the SEO functionality your Marketing / SEO person will need. With that said, many websites will only need minor textual changes throughout your websites' code and your SEO person should be able to handle them. Many people react and act too quickly on this subject. A true marketing person will need to track your website for a year or longer. Changes to your on-site SEO should happen during this time and your website should continue to grow from what it starts as. This phase never stops, unless a website is dead.
With that being said, it is possible to find those that have the ability to do all of this for you. Front-End design today has completely changed and the ability to obtain works that developers have spent months on, for pennies on the dollar are widely available. These are often good starting places to give you a foundation to build from. Doing so will cut down on the number of hours you need to hire someone and the number of people you need to hire.