Standard feed forward deep net: like the ones you used at university but with a few important features. One, you can stack layers on top of each other then train the whole network with back propagation. The nonlinearity you use can be important (rectified nonlinear units--Max(0,x) are popular now). Regularization can be important (with dropout being a popular method). Pre-processing your data can also be important (eg scaling the inputs, subtracting the mean, ...). How you initialize your weights is important. Using tricks like momentum and learning rate decay are important too.
Autoencoder: A standard feed forward deep net that tries to output an "uncorrupted" sample of a "corrupted" input. So basically, take an image, add noise, ask it to output the denoised image. Why you ask? Well, in doing this, the hidden units (the layers in between the input and output) tend to discover new features which can then be used elsewhere in machine learning pipelines. The big win here is automated feature engineering.
RBM: Restricted boltzman machine, sort of like an autoencoder but not. I'm not convinced one is better than the other, but there are definitely differences.
Recurrent neural networks: Like standard feed forward deep nets but extended to time series problems. The hidden units at each timestep feed in as additional input to the next timestep (along with the new input). So basically, each step it gets this as input: [whatever came before described as hidden units] + [input at T]. These are my personal favorite and currently hold state of the art in speech recognition, at least academically.
Convolutional neural networks: State of the art in vision, this one is kind of hard to explain quickly, but you can think of it kind of like this: I take some image where I want to recognize objects in it. Then I train a bunch of neural nets that represent different aspects of an image (one net might look for vertical lines, another might look for horizontal. It figures out what it's looking for automatically and distributes what it thinks are important across these little networks. You can think of this as "feature detection") You then take a window on the image (400x400 image, maybe you've got a 20x20 window) which you then slide across the image and check each section for the presence of whatever the net is looking for (vertical/horizontal lines, etc.) There's several layers of this (specifically a convolution operation followed by a pooling operation) before the result is fed to a standard fully connected feed forward net which then outputs a prediction.
While lower layers look for low level aspects of an image, each layer progressively looks for higher and higher level phenomena--for instance, when you hear about "the cat neuron" it comes from probing a neuron in the higher levels of a convolutional net and finding that pictures of cats happen to turn this neuron "on" while pictures of anything else don't. What is "high level" exactly? In this case, it really just means composed from "lower level" components... Some also say "higher level = more abstract" but I don't know this is quite the case--abstract means something different to me.
An important consideration is that the conv net architecture allows you to cut down on the total number of parameters (good when you've got something high dimensional like image data) and it also takes advantage of the fact that you can make certain assumptions about images and how objects move/appear in space basically.
Conv nets are probably the most difficult architecture to grasp and my explanation is extremely high level.
Ha, that was a bit longer than I had hoped, but I hope it's helpful to someone...