This is quite a common question that people in web design or development get asked a lot, and something a lot of people are unsure about.
It's actually something I studied in depth whilst at uni.
The short answer is that design is literally just how it looks, and development is how it is build and how it works.
In general the answer to this question varies for different companies, depending on its size, target audiences etc.
Some companies may have a design team and a development team, whereas freelancers will generally do both design and development. The role of designers and developers is largely dependant on the structure of the team. If there is a design team and a development team, then the chances are that the design team will use something like Photoshop to draw up the design and pass it onto a developer to build it as a website.
Here at Innovation & Development Centre in Torquay, we actually have one Design team and one Development team, but as we are all working together in one room, the teams are combined. This works particularly well because we all have knowledge of both design and development, each of us with a different strength.
The designers will generally design websites in Photoshop first, before then creating the website based on that design, using the WebBoss system. When more complex functionality is required, it may be referred to a developer to implement this.
The developers, other than helping with the technical side of websites, focus on improving and developing the WebBoss system with new features and modules.
Here's another simple infographic to explain the differences. 
This isn't true for all designers and developers.
For example, all our designers know the basic coding languages like HTML and CSS and although I'm more of a developer, my degree was Graphic Design before I specialised in Web Design and moved on to development.
However, this does give a nice general overview.
So if you want to change something on your website, ask yourself these questions.
Does it affect how the website works? If so, it's probably development.
Is it more about how the website looks? If so, it's probably design.
If you're still unsure and have a question about your website, we can direct your query to the correct team to find the answer you're looking for.