Artificial neural networks (ANNs) are mathematical constructs, originally designed to approximate biological neurons. Each "neuron" is a relatively simple element --- for example, summing its inputs and applying a threshold to the result, to determine the output of that "neuron".

Several decades of research went into discovering how to build network architectures using these mathematical constructs, and how to automatically set the weighting on each of the connections between the neurons to perform a wide range of tasks. For example, ANNs can do things like recognition of hand-written digits.

A "biological neural network" would refer to any group of connected biological nerve cells. Your brain is a biological neural network, so is a number of neurons grown together in a dish so that they form synaptic connections. The term "biological neural network" is not very precise; it doesn't define a particular biological structure.

In the same way, an ANN can mean any of a large number of mathematical "neuron"-like constructs, connected in any of a large number of ways, to perform any of a large number of tasks.

I'm studying computer science at KIT (Karlsruhe Institute of Technology, Germany) specializing in Machine Learning and a minor in Mathematics. I am not a biologist.

An artificial neural network is basically a mathematical function. It is built from simple functions which have parameters (numbers) which get adjusted (learned). One example of such a function is:

$$varphi(x_1, dots, x_n) = frac{1}{1+e^{sum_{i=1}^n w_i x_i}} text{ with } w_i in mathbb{R}, x_i in mathbb{R}$$

As you can see, there are parameter $w_i$ which can be set to any number. The $x_i$ are the inputs. But those inputs themselves can be the output of another function $varphi$.

A common way to visualize those simple functions is

There are thousands of those simple functions (also called "neurons") which build an artificial neural network. This results in a single function which has millions of parameters in the biggest networks. Neural networks use relatively simple topologies which can often be expressed in a tree structure:

Every single circle is one neuron (also called "perceptron" in some cases).

You have lots of training data, train once a long time, then use it often. A long time often means several days.

In an artificial neural network (ANN), the signal is send synchronously.

As far as I know, the human brain isn't fully understood by now. From what I've heard, the basic building blocks are also called neurons. They look like that:

The idea is that axon terminals of one neuron, which are closer to the axon hillock of another neuron have more influence on the output of the signal. But the relationship is different. It is a process which is much better described by differential equations, not something simple as the equation for artificial neurons above. The neurons send their signals asynchronously. According to Wikipedia, humans have 86 000 000 000 neurons. They are certainly restricted in how they can be connected, but there are also certainly many billions of parameters which can be adjusted in "standard human brains".

By now, we don't know how humans really (exactly, in all details) learn. This is certainly a key difference. But we don't learn once for a long time and stop learning then. We are continuously learning very short times.

See also: On-line Recognition of Handwritten Mathematical Symbols, Chapter 4.3 for more details.

TL;DR and some more facts

• Signal transport and processing: The human brain works asynchronously, ANNs work synchronously.
• Parameter count: Humans have many billions (businessinsider.de writes "1,000 trillion") of adjustable parameters / 86Bn neurons (source). Even the most complicated ANNs only have several million learnable parameters.
• Training algorithm: ANNs use Gradient Descent for learning. Human brains use something different (but we don't know what)
• Processing speed: Single biological neurons are slow, while standard neurons in ANNs are fast. (Yes, I did not confuse the two. The artificial ones are faster. And I ignore spiking networks as I didn't see a single example where they are actually useful)
• Topology: Biological neural networks have complicated topologies, while ANNs are often in a tree structure (I am aware of recurrent networks - they are still pretty simple from their structure)
• Power consumption: Biological neural networks use very little power compared to artificial networks
• Biological networks usually don't stop / start learning. ANNs have different fitting (train) and prediction (evaluate) phases.
• Field of application: ANNs are specialized. They can only perform one task. They might be perfect at playing chess, but they fail at playing go (or vice versa). Biological neural networks can learn completely new tasks.
• ANNs need a target function. You need to tell them what is good and what is bad. I'm not sure if you can compare this to e.g. pain in biological networks.