Use Colored Line Art.
This was rather difficult in traditional animation, but with 2D computer animation it's easy to create outlines in colors other than standard black. You might want to use a light brown for the line art around flesh-toned areas, or a darker blue outline on a pale blue shirt. This creates a softer, more blended look for the animation, so that it becomes a more seamless part of the background and creates an almost portrait-like appearance. For an example, take a look at the end result of my lesson on retracing detailed line art in Flash (and the part on creating tone via different line colors). In almost every area, I used colored lines to blend the art together more smoothly.
Play With Angles and Zoom to Dramatic Effect.
A lot of animations use scene compositions that make them look like a side-scrolling video game. That's not necessarily bad, but it doesn't really stand out. There's no real reason to conform to that style all the time, and when you make clever use of angles, perspective, and zoom, you can create dramatic effects that heighten the mood of your animated scenes. For example, when a character is delivering a dramatic monologue, use a front view - but with one half of the character's face cut off by the edge of the screen, leaving the rest filled with black (or even with another animation depicting what they're talking about). It creates a stark, grim feel, with the character's one eye focused on you. Another way is to zoom in only on their mouth, so that all emotion has to be portrayed by the curvature of the mouth and tone of voice. You can use tilted angles to display the confusion of the character observing events, sharp high angles to create surrealism, or exaggerated perspective from low angles to make someone appear to loom, larger than life. The only limit to what you can do here is your own imagination.
Use 2.5D animation tricks.
2.5D animation crosses that line between 2D and 3D animation, and creates an eye-catching sense of depth. This can involve anything from establishing a 3D scene perspective through use of character shadows to creating a false perspective and sense of three-dimensionality using little tricks that make your characters and objects appear to occupy a 3D space instead of a flat page (such as changing perspective on a head turn so that the character's head really does appear to be spherical, instead of just round).
Be Unorthodox In Your Character Designs.
You don't have to use perfect proportions, or classical toon style. Do something different. Character designs are as much a part of what makes an animation stand out as anything else, and if your characters are unique, your animation will stick out in people's minds. One striking example is 2D from the band Gorillaz; his empty, hollow eye sockets are eerie and entirely memorable, and despite not having eyeballs he's still animated with a great deal of emotion. You can also look at styles such as those in the show Winx Club: long and leggy and attenuated, mimicking fashion design sketches. It's the ones that break the norms and defy convention that make you stop and take a second look - so don't be afraid to do something a little different from the textbook.
Take Your Extremes to New Extremes.
Animation is all about extremes - using squash and stretch, exaggeration, anticipation, attraction, and other tricks to draw viewers into an experience that's more real than reality. Animations have to go big or go home; if they try to convey emotion and action using realistic expressions and motions, they end up falling flat for numerous reasons, one being that they lack the body language and other cues that real people have to support their expressions and motions. While extremes are standard in animation, you can take yours to the next level and really go all-out in exaggerating your animation until it's like a sledgehammer to the face. Ever seen FLCL? Yeah, that one will smash you upside the head with its extremes and then keep kicking you until you can't get up.
We're no longer confined to strictly 2D or strictly 3D. You can mix mediums and methods in as many ways as you want, whether you're making a 2D character move through a wholly 3D background or mapping 2D art onto 3D animated shapes. You can even mix mediums by joining traditionally hand-painted cel animations with Flash animation work, or bringing in detailed artwork from Photoshop to use little tricks to animate it. Combine your expertise in unique ways to make your style truly distinctive.
There are tons of other ways to really make your animation and your style stand out. The biggest thing? Let yourself think different. Don't just copy what you see other people doing. Try new things, and if they bomb, try something else. These tips are only meant to give you ideas, and act as a springboard to push you in the right direction. Tilt your world upside down, see how things look from there...and then animate it in a way that people will never forget.