The short answer is, a lot. Colour is incredibly important when it comes to the design of a website. It is one of the most powerful ways of subconsciously letting a visitor know what you and your product are about. It is an extension of you, your untold hours of hard work and as such it warrants a lot more thought then you may initially think. Look at Coca-Cola as an example, while it is not confirmed that they are behind the full re-brand, they certainly popularised the traditional red image of Santa to fit their branding/colour scheme when they commissioned Swedish painter Haddon Sundblom to paint several versions for Christmas adverts. This is now the version of Santa most people have in their homes and in their hearts at what is considered most people's (I include myself here) favourite time of the year. Still have doubts over how powerful a colour can be to a brand/website? Read on.
Colour has always been one of the most fascinating aspects of website design to me, as I mentioned it can be a very powerful way of evoking instant emotional connections. Using a green instead of a purple for example can change the entire dynamic of a website, content will read differently, imagery will need to be shifted and people will expect different things from the website and the brand/business as a result. I find it fascinating to learn about the associations people have with colours and how brands manipulate this, for example, Starbucks moved from using brown to green to subconsciously portray that they are fresh and clean. Apple have managed to become the masters of branding and they have done so by embracing white, making people buy into their "we are the future" mantra. Simple, yet highly effective as Apple are now bordering on becoming a religion.
I find that most designers have very differing opinions on what is too much or too little colour within a project. It has been a troublesome road within web design. Back in the day, it seemed to be the norm to throw as many colours as possible at a website just to show what they were capable of. Compare this with today's one-page monochromatic design where huge imagery is the norm. Colour is an afterthought it seems. I like to work as much of a brand's colour into a site as I can, not so much that it becomes over powering but enough to re-inforce the colouring and therefore the brand within the browser's mind. Simple items like buttons, hyperlinks, font awesome icons and moz selections can all be adhered to a client's colouring without overpowering a site circa 2000's style.
My own personal favourite colour is red which is why I used it within my own site and branding. It is the colour I have long identified with and have always been drawn to. Apparently since red is the colour of "blood and fire, is associated with meanings of love, passion, desire, heat, longing, lust, sexuality, sensitivity, romance, joy, strength, leadership, courage, vigor, willpower, rage, anger, danger, malice, wrath, stress, action, vibrance, radiance, and determination." Santa, Coca-Cola and I have a lot in common.