Neural networks are widely used in unsupervised learning in order to learn better representations of the input data. For example, given a set of text documents, NN can learn a mapping from document to real-valued vector in such a way that resulting vectors are similar for documents with similar content, i.e. distance preserving. This can be achieved using, for example, auto-encoders - a model that is trained to reconstruct the original vector from a smaller representation (hidden layer activations) with reconstruction error (distance from the ID function) as cost function. This process doesn't give you clusters, but it creates meaningful representations that can be used for clustering. You could, for instance, run a clustering algorithm on the hidden layer's activations.
Clustering: There are a number of different NN architectures specifically designed for clustering. The most widely known is probably self organizing maps. A SOM is a NN that has a set of neurons connected to form a topological grid (usually rectangular). When some pattern is presented to an SOM, the neuron with closest weight vector is considered a winner and its weights are adapted to the pattern, as well as the weights of its neighbourhood. In this way an SOM naturally finds data clusters. A somewhat related algorithm is growing neural gas (it is not limited to predefined number of neurons).
Another approach is Adaptive Resonance Theory where we have two layers: "comparison field" and "recognition field". Recognition field also determines the best match (neuron) to the vector transferred from the comparison field and also have lateral inhibitory connections. Implementation details and exact equations can readily found by googling the names of these models, so I won't put them here.