In theory you can upload someone's mind onto a computer, allowing them to live forever as a digital form of consciousness, just like in the Johnny Depp film Transcendence.
But it's not just science fiction. Sure, scientists aren't anywhere near close to achieving such feat with humans (and even if they could, the ethics would be pretty fraught), but now an international team of researchers have managed to do just that with the roundworm Caenorhabditis elegans.
Uploading an animal, even one as simple as c. elegans would be very impressive. Unfortunately, we're not there yet. What the people working on Open Worm have done instead is to build a working robot based on the c. elegans and show that it can do some things that the worm can do.
The c. elegans nematode has only 302 neurons, and each nematode has the same fixed pattern. We've known this pattern, or connectome, since 1986.  In a simple model, each neuron has a threshold and will fire if the weighted sum of its inputs is greater than that threshold. Which means knowing the connections isn't enough: we also need to know the weights and thresholds. Unfortunately, we haven't figured out a way to read these values off of real worms. Suzuki et. al. (2005)  ran a genetic algorithm to learn values for these parameters that would give a somewhat realistic worm and showed various wormlike behaviors in software. The recent stories about the Open Worm project have been for them doing something similar in hardware. 
To see why this isn't enough, consider that nematodes are capable of learning. Sasakura and Mori (2013)  provide a reasonable overview. For example, nematodes can learn that a certain temperature indicates food, and then seek out that temperature. They don't do this by growing new neurons or connections, they have to be updating their connection weights. All the existing worm simulations treat weights as fixed, which means they can't learn. They also don't read weights off of any individual worm, which means we can't talk about any specific worm as being uploaded.
If this doesn't count as uploading a worm, however, what would? Consider an experiment where someone trains one group of worms to respond to stimulus one way and another group to respond the other way. Both groups are then scanned and simulated on the computer. If the simulated worms responded to simulated stimulus the same way their physical versions had, that would be good progress. Additionally you would want to demonstrate that similar learning was possible in the simulated environment.
(In a 2011 post on what progress with nematodes might tell us about uploading humans I looked at some of this research before. Since then not much has changed with nematode simulation. Moore's law looks to be doing much worse in 2014 than it did in 2011, however, which makes the prospects for whole brain emulation substantially worse.)
I also posted this on my blog.
 The Structure of the Nervous System of the Nematode Caenorhabditis elegans, White et. al. (1986).
 A Model of Motor Control of the Nematode C. Elegans With Neuronal Circuits, Suzuki et. al. (2005).
 It looks like instead of learning weights Busbice just set them all to +1 (excitatory) and -1 (inhibitory). It's not clear to me how they knew which connections were which; my best guess is that they're using the "what happens to work" details from . Their full writeup is .
 The Robotic Worm, Busbice (2014).
 Behavioral Plasticity, Learning, and Memory in C. Elegans, Sasakura and Mori (2013).