The default example is classification of a circle of one class surrounded by a donut of another. There are two features x_1 and x_2 (this is the "raw data").
One solution to this problem is to use a single layer and a single neuron but engineer features manually. These manually engineered features are x_1*x_2, x_1^2,x_2^2, sin(x_1) and sin(x_2). Here's a link to this model (long url)[1].
This model performs very well at learning to classify the data just by combining these manual features with a single neuron. The problem is a human needs to figure out these features. Try removing some and observe the different performance given different manual features. You'll see how important it is to engineer the correct ones.
Alternatively you can have 2 layers of 4 neurons [2]. In nearly the exact number of iterations this network also learns to classify the data correctly. This is because the non-linear interactions between neurons are actually transforming the inputs the appropriate ways. That is to say the networks is learning to engineer the features itself. Try removing layers/nodes and you'll find that a simpler network will have a harder and harder time at this.
I recommend playing around with the various tradeoff between manually engineered features and network complexity. The interesting thing you will observe is that in some cases the manual features are much faster to learn a simplier model than the network. The big issues comes up when we can't simply "see" the problem in 2d so we have no idea what features may and may not be useful.
[0] http://playground.tensorflow.org/
[1] http://playground.tensorflow.org/#activation=tanh&batchSize=...
[2]. http://playground.tensorflow.org/#activation=tanh&batchSize=...