Why doesn't [CSS feature] work in [browser] but works in others?

Though it is not always the case, one of the most common reasons why a property like transition or animation works on some browsers and not others is because of vendor prefixes.

What are vendor prefixes?

At the time version 4 of Firefox was introduced, the CSS transition module specification was a Working Draft. Before a spec is finalized (in practice, this is when it reaches Candidate Recommendation), browser vendors add vendor prefixes to properties, values, and @-rules to prevent compatibility problems in case the spec changes.

Vendor prefixes are exactly what their name describes - a vendor-specific (vendor meaning a company who develops a browser) prefix of a property or value. They are often implemented in a specific way for each browser because the property or value is still in one of the many experimental phases before the Candidate Recommendation stage, which is the stage where they are considered implementation-ready.

The most common ones are -moz- (Mozilla Firefox), -webkit- (Chrome, Safari, etc.), and -ms- (Microsoft Internet Explorer), but there are more.

When do I need to use them?

That depends completely on what browsers you're looking to serve, what properties and values you're using, and at what point in time you are developing your website. There are sites that try to keep a current list but they are not always accurate or kept up-to-date.

Following are some of the most commonly prefixed properties and values. If your project does not supporting the browsers mentioned in the comment to the right of the property, then there is no need to include it in your CSS.

Transitions

An unprefixed property sometimes has prefixed equivalents, such as -webkit-transition.

In order to get full possible browser support, the following is necessary:

.foo {
    -webkit-transition: <transition shorthand value>; /* Safari 3.1-6, Chrome 1-25, Old Android browser, Old Mobile Safari, Blackberry browser */
    -moz-transition: <transition shorthand value>;    /* Firefox 4-15 */
    -o-transition: <transition shorthand value>;      /* Old opera */
    transition: <transition shorthand value>;         /* Modern browsers */
}

Note that an -ms- prefix exists for transition, however it was only implemented by pre-release versions of IE10 which are no longer functional, and it is therefore never needed. It is implemented unprefixed in IE10 RTM and newer.

Transforms

.foo {
    -webkit-transform: <transform-list>; /* Chrome 21-35, Safari, iOS Safari, Opera 22, many mobile browsers */
    -ms-transform: <transform-list>;     /* IE9 */
    transform: <transform-list>;
}

Animations

Animations need to have the property prefixed and the corresponding keyframes prefixed, like so:

.foo {
  -webkit-animation: bar; /* Safari 4+ */
  -moz-animation: bar;    /* Fx 5+ */
  -o-animation: bar;      /* Opera 12+ */
  animation: bar;         /* IE 10+, Fx 16+ */
}

@-webkit-keyframes bar { /* Keyframes syntax */ }
@-moz-keyframes bar { /* Keyframes syntax */ }
@-o-keyframes bar { /* Keyframes syntax */ }
@keyframes bar { /* Keyframes syntax */ }

Flexbox

Values can also be prefixed, as in the case of flexbox. Note: For maximum browser compatibility, flexbox-specific properties like ordinal-group, flex-flow, flex-direction, order, box-orient, etc. need to be prefixed in some browsers in addition to the following:

.foo {
    display: -webkit-box;  /* OLD - iOS 6-, Safari 3.1-6 */
    display: -moz-box;     /* OLD - Firefox 19- (buggy but mostly works) */
    display: -ms-flexbox;  /* TWEENER - IE 10 */
    display: -webkit-flex; /* NEW - Chrome */
    display: flex;         /* NEW, Spec - Opera 12.1, Firefox 20+ */

    -webkit-box-flex: <flex shorthand value>;
    -moz-box-flex: <flex shorthand value>;
    -webkit-flex: <flex shorthand value>;
    -ms-flex: <flex shorthand value>;
    flex: <flex shorthand value>;
}

Calc

.foo {
    width: -webkit-calc(<mathematical expression>); /* Chrome 21, Safari 6, Blackberry browser */
    width: -moz-calc(<mathematical expression>);    /* Firefox <16 */
    width: calc(<mathematical expression>);         /* Modern browsers */
}

Gradients

See CSS Gradients on CSS-Tricks for more information.

.foo {
    background-color: <color>; /* Fallback (could use .jpg/.png alternatively) */
    background-image: url(bar.svg); /* SVG fallback for IE 9 (could be data URI, or could use filter) */  
    background-image: -webkit-gradient(linear, left top, right top, from(<color-stop>), to(<color-stop>)); /* Safari 4, Chrome 1-9, iOS 3.2-4.3, Android 2.1-3.0 */  
    background-image: -webkit-linear-gradient(left, <color-stop>, <color-stop>); /* Safari 5.1, iOS 5.0-6.1, Chrome 10-25, Android 4.0-4.3 */  
    background-image: -moz-linear-gradient(left, <color-stop>, <color-stop>); /* Firefox 3.6 - 15 */
    background-image: -o-linear-gradient(left, <color-stop>, <color-stop>); /* Opera 11.1 - 12 */
    background-image: linear-gradient(to right, <color-stop>, <color-stop>); /* Opera 15+, Chrome 25+, IE 10+, Firefox 16+, Safari 6.1+, iOS 7+, Android 4.4+ */
}

Note that left and to right represent the same direction, left-to-right, and therefore left and to left point opposite ways. See this answer for some background info.

Border-radius (Not needed in most cases)

.foo {
    -webkit-border-radius: <length | percentage>; /* or iOS 3.2 */
    -moz-border-radius: <length | percentage>;    /* Firefox 3.6 and lower */
    border-radius: <length | percentage>;
}

Box shadow (Not needed in most cases)

.foo {
    -webkit-box-shadow: <box-shadow shorthand value>; /* iOS 4.3 and Safari 5.0 */
    -moz-box-shadow: <box-shadow shorthand value>;    /* Firefox 3.6 and lower */
    box-shadow: <box-shadow shorthand value>;
}

How can they be implemented with JavaScript?

To access prefixed attributes and events in JavaScript, use the camelCase equivalent of the CSS prefix. This is true for event listeners like foo.addEventListener('webkitAnimationIteration', bar ) as well (foo being a DOM object, like document.getElementsById('foo')).

foo.style.webkitAnimation = '<animation shorthand value>';
foo.style.mozAnimation = '<animation shorthand value>';
foo.style.oAnimation = '<animation shorthand value>';

Prefixing tools

Online prefixers can be helpful but are not always reliable. Always make sure to test your project on the devices you wish to support to make sure that each has the appropriate prefix included.

CSS Pre-processor functions:

SASS & SCSS properties prefixer
LESS properties prefixer

JavaScript prefixer functions:

-prefix-free for CSS properties and values
Event prefixer

See also: Why do browsers create vendor prefixes for CSS properties?