What does the activation of a neuron mean?
The activation of a neuron is mathematically nothing but a function of its input. Consider a neural network with one hidden layer and one input vector $\mathbf{x}$. The input $a_j$ of neuron $j$ can be written as:  

$a_j = w_j^T \mathbf{x} + b_j$. 

The activation of neuron $j$ is then a transformation $g: \mathbf{R} \rightarrow \mathbf{R}$ of the input. For exmaple, one can use the sigmoid activiation function

$sig(a_j) = \dfrac{1}{1 + \exp(-a_j)} $

The weights $w$ are chosen to minimize a loss function and not for the sake of their interpretation. Nevertheless, one can try to interpret the activation of a neuron as internal representation of the input. One of the nicest examples I've seen comes from the Rummelhart et al (1986), Figure 4. In that paper, two family trees were fed into a neural network. The family trees represented an Italian and an English speaking family comprising three generations. Among other things, when activating one name of the tree, the neuronal activities represented whether the person was from the English or Italian tree and which generation the person embodied.