What are linearly dependent vectors like? How are they different from linearly independent vectors?  

While the above answer is correct, I think some intuitive understanding might help you, too. 
Consider $\mathbb{R}^2$. Take the vectors $u= (1,1)$ and $v=(-2,-2)$. These vectors are called linearly dependent because they are equal up to a linear transformation, i.e. $v = (-2) \cdot u$.
This generalizes to $\mathbb{R}^n$ and more generally any kind of vector space using the definition given by Amstell above.
