How to compare the output of a neural network with his target?
The easy answer: Take the predicted class to be the argmax of your output vector. 

The longer answer: Since you're doing multiclass classification you should probably be using softmax output units (if I had to guess, I would guess you're using sigmoid output units). If $\mathbf{x} \in \mathbb{R}^n$ is the input to your output units then the softmax for the $i$th output unit is

$$ y_i = \frac{\exp(x_i)}{\sum_{j}\exp(x_j)} $$

This output function is nice for a number of reasons.


since your output vector sums to 1, you can interpret it probabilistically.
the derivative is particularly nice in combination with a log loss function.


Note: the further information I've provided may seen redundant since clearly

$$ \text{argmax}\{\mathbf{x}\} = \text{argmax}\{\mathbf{\sigma(x)}\} = \text{argmax}\{\mathbf{\text{softmax}(x)}\}, $$

where $\sigma$ is the elementwise logistic sigmoid function. The problem with just taking the argmax of what you have is you will likely not be optimizing the function you are actually interested in internally. 

On the other hand the choice of loss function for classification isn't as cut and dry as I've made it sound.