An industrial design or product design firm or agency is a consulting company that works with a company to productise their ideas. Traditionally, product design companies work on tangible, physical products. Companies that develop software are typically not called product design agencies or consulting companies. Instead, they might brand themselves a software development agency, a staff extension agency, an offshore development shop, a web design and development firm or some such.
Of US based product design agencies, IDEO is the most well known example. There are lots of design firms that specialize in different sectors, from robotics to healthcare to consumer electronics and more.
These companies are a great fit for a startup (or for that matter, a new product team within a big company) that has a strong R&D team but no in house expertise in bringing a product to market. Oftentimes the client will approach a design firm with an idea, a working breadboard or engineering prototype, and a vision for where they would like to take this product. The design firm would then put together a project team comprising industrial designers, mechanical engineers, and electrical and electronics engineers as needed to develop the product for them.
Special thanks: Martin Trust Centre, MIT