Responsive web design is to put it simply, about planning out the design to "respond" to the change in environment, size of the screen, browser type, or application. A good designer will plan not only for a large desktop, but also tablets, and mobile devices as many users will experience your site on multiple sizes. Developers use media queries, column layouts, responsive images, and font sizes adjusting to improve the user experience on mobile phones, tablets, and desktops so that the web page looks it's best on any (*or almost any -- I'll get to that in a minute) screen size.
Ask! Seriously, have a conversation with anyone you're looking to hire to handle the design of your site if responsive design is part of the plan. If there is any pause than I recommend moving on as this is critical to a good user experience, which in turn affects how well your site will do in searches.
Look for a nice reflow of the content you have on your site, as well as images resizing appropriately for the mobile devices you test it on. It shouldn't have lots of overly long scrolls, super huge images, or buttons too small to use on a small screen. The responsive layout should still feel like your site, but in a smoother, sleeker, smaller screen layout. Font sizes should also adjust accordingly so the headers of the page are not too big, and the content should not be too small. Grids for content and imagery will usually pare down to one column layout.
Absolutely! When you're working with your designer and developer on the site build, there should be multiple rounds of reviews and they should include various screen sizes and devices. Media queries will be created by your developer to distinguish between a desktop device and a mobile phone. Make sure to have them review where their breakpoints are for the site and if you have analytics data available see what percentage of your users typically use a certain device width or screen size and optimize the pages to make those user experiences shine! You want them to come back to the site often and like using your web page, so make it a really good web page for them.
Make sure pop ups resize appropriately, and that buttons, links and menu toggles are accessible to a finger tip size, otherwise they will prove frustrating for the user.
I have also been really pushing clients to consider accessibility when designing a website as well. This applies to all forms of the website, not just mobile and responsive web design. Color choices and contrast should also be considered. You do not want to put a light gray button on a white background. This is difficult to see for many users, and phones they may have contrast turned down. Try to make sure content is still readable, and not too small. Make headlines not to big and thus difficult to read and not mobile friendly.
I typically get a site to the point that 98% of the users are having a good experience -- this isn't to say that things will always be perfect. I had a client not long ago moved her browser window on her desktop to be so short, that you barely could fit the header, much less the content. She was upset because this obscure proportion she had chosen didn't look good. The device width initial scale will have some proportions considered in the result. Tablets and other devices as well will have a typical proportion to them. But obscure ones may be missed, so if you feel those are important, communicate that to your designer and developer ahead of time so they can plan accordingly. There may be extra time billed if some odd device few people use is important to you, but if it is, please say so!
A Girl Creative Design Studio, LLC | Denver, CO