How does a convolutional ply differ from an ordinary convolutional network?
It seems that a convolutional ply is exactly the same as an ordinary convolutional layer. From their paper, they argue that the term "CNN layer" usually refers to a convolutional layer followed by a pooling layer. In an attempt to reduce confusion, they name the convolutional part a "convolution ply" and a the pooling part a "pooling ply":


  In CNN terminology, a pair of convolution and pooling layers in Fig. 2 in succession is usually referred to as one CNN “layer.” A deep CNN thus consists of two or more of these pairs in succession. To avoid confusion, we will refer to convolution and pooling layers as convolution and pooling plies, respectively.


Ironically, this has increased the confusion, leading to this post. At the time I suppose it was not common to have several convolutional layers in a row before a pooling layer, but we see this often in modern architectures. 

To answer your other question about the network structure; they state the structure of the network that they use in the Experiments section (Section V-B). To hopefully reduce confusion, I've replaced the word "ply" with "layer":


  In these experiments we used one convolution [layer], one pooling [layer] and two fully connected hidden layers on the top. The fully connected layers had 1000 units in each. The convolution and pooling parameters were: pooling size of 6, shift size of 2, filter size of 8, 150 feature maps for FWS, and 80 feature maps per frequency band for LWS.