As stated by the other answers, you need to provide @font-face directives for all the typefaces you want to us on your site. You currently have:
@font-face {
font-family: 'Avenir Next';
font-weight: normal;
font-style: normal;
font-variant:normal;
src: url('./fonts/AvenirNext-Regular/AvenirNext-Regular.eot?') format('eot'),
url('./fonts/AvenirNext-Regular/AvenirNext-Regular.otf') format('opentype'),
url('./fonts/AvenirNext-Regular/AvenirNext-Regular.woff') format('woff'),
url('./fonts/AvenirNext-Regular/AvenirNext-Regular.ttf') format('truetype'),
url('./fonts/AvenirNext-Regular/AvenirNext-Regular.svg#AvenirNext-Regular') format('svg');
}
So just for clarity, this is what you need to include all the different styles (regular, bold, italicized, bold and italicized):
@font-face {
font-family: 'Avenir Next';
font-weight: normal;
font-style: normal;
font-variant:normal;
src: url('./fonts/AvenirNext-Regular/AvenirNext-Regular.eot?#iefix') format('eot'),
url('./fonts/AvenirNext-Regular/AvenirNext-Regular.otf') format('opentype'),
url('./fonts/AvenirNext-Regular/AvenirNext-Regular.woff') format('woff'),
url('./fonts/AvenirNext-Regular/AvenirNext-Regular.ttf') format('truetype'),
url('./fonts/AvenirNext-Regular/AvenirNext-Regular.svg#AvenirNext-Regular') format('svg');
}
@font-face {
font-family: 'Avenir Next Bold';
font-weight: bold;
font-style: normal;
font-variant:normal;
src: url('./fonts/AvenirNext-Bold/AvenirNext-Bold.eot?#iefix') format('eot'),
url('./fonts/AvenirNext-Bold/AvenirNext-Bold.otf') format('opentype'),
url('./fonts/AvenirNext-Bold/AvenirNext-Bold.woff') format('woff'),
url('./fonts/AvenirNext-Bold/AvenirNext-Bold.ttf') format('truetype'),
url('./fonts/AvenirNext-Bold/AvenirNext-Bold.svg#AvenirNext-Bold') format('svg');
}
@font-face {
font-family: 'Avenir Next Italic';
font-weight: normal;
font-style: italic;
font-variant:normal;
src: url('./fonts/AvenirNext-Italic/AvenirNext-Italic.eot?#iefix') format('eot'),
url('./fonts/AvenirNext-Italic/AvenirNext-Italic.otf') format('opentype'),
url('./fonts/AvenirNext-Italic/AvenirNext-Italic.woff') format('woff'),
url('./fonts/AvenirNext-Italic/AvenirNext-Italic.ttf') format('truetype'),
url('./fonts/AvenirNext-Italic/AvenirNext-Italic.svg#AvenirNext-Italic') format('svg');
}
@font-face {
font-family: 'Avenir Next Bold Italic';
font-weight: bold;
font-style: italic;
font-variant:normal;
src: url('./fonts/AvenirNext-Bold-Italic/AvenirNext-Bold-Italic.eot?#iefix') format('eot'),
url('./fonts/AvenirNext-Italic/AvenirNext-Bold-Italic.otf') format('opentype'),
url('./fonts/AvenirNext-Italic/AvenirNext-Bold-Italic.woff') format('woff'),
url('./fonts/AvenirNext-Italic/AvenirNext-Bold-Italic.ttf') format('truetype'),
url('./fonts/AvenirNext-Italic/AvenirNext-Bold-Italic.svg#AvenirNext-Bold-Italic') format('svg');
}
This is assuming the actual font names of course. Once this is in place, you can use the fonts as per @rubo123's answer. I added the #iefix to terminate the query string as explained in this question: css - How does ?#iefix solve web fonts loading in ie6-8?