What are linearly dependent vectors like? How are they different from linearly independent vectors?  
 A: Linear dependent vectors $X_{1},X_{2},...,X_{n}$ arise when there exists a scalar $c_{1},c_{2},...,c_{n}$ such that :
$\sum_{i=1}^{n}c_{i}X_{i}=0$
If the determinant is 0, then the vectors are linearly dependent:
$$
\begin{vmatrix}
\ x_{11} & x_{12} & ...& x_{1n} \\
\ x_{21} & x_{22} & ... & x_{2n} \\
\ \vdots & \vdots & \ddots & \vdots \\
\ x_{n1} & x_{n2} & \cdots & x_{nn}
\end{vmatrix}
=0\notag
$$
You can think of dependency as vectors having some relationship to each other (e.g. similar variables).  If there is no scalar that exists then the vectors are linearly independent.
Example
$x_{1} =
\begin{pmatrix}
\ 1 \\
\ 1 \\
\ 1
\end{pmatrix}$
$x_{2} =
\begin{pmatrix}
\ 1 \\
\ -1 \\
\ 2 \\
\end{pmatrix}$
and $x_{3} =
\begin{pmatrix}
\ 3 \\
\ 1 \\
\ 4 \\
\end{pmatrix}$
These vectors are linearly dependent since $2x_{1} + x_{2} - x_{3} = 0$
A: 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.
