Well, the question is in the title, no explanation need.
I've had a little encounter with this question in the past few months so I'm updating here...
The overall answer is 'really a lot about some things, but not as much as you'd like to think about others.' There is a scientific genome interpretation 'contest' that has been going on for the past few years called CAGI (Critical Assessment of Genome Interpretation). This is meant to be a cutting edge set of challenges and its worth looking them over. Last year there was in particular one challenge - answering questions about ten individuals given only a list of traits and their genome sequences.
It was not so easy it turns out - simply looking up variants and cross referencing them to the literature led to poor predictions. Glaucoma, asthma, migrane, irritable bowel syndrome, color blindness, lupus, lactose intolerance are examples from a list of 40-odd contest questions. If you register onto the site you can get some of the results or there is a paper reporting the results, only four labs tried the challenge and the accuracies were sometimes not great, topping off with Rachel Karchin's lab with an AUC of 90%. Even some of the phenotypes you think are easy are not a simple lookup. Genetics is not as pre-determined as we think.
While we are good at inferring our history and geneology on the other hand. A reasonable example of this is the 23andme analysis. They have a lot less information about you than a complete genome sequence, but they do seem to make the most of the 0.0001% of the genome that they do have.
They have a nice lookup of some of the hereditary disease data that is available. "Increased risk of skin cancer" or for Alzheimers are only chances and its never clear how much your actual lifestyle has impact on 44 outcomes, but its there.
What they also have is your ancestral analysis which is crazy interesting. it shows where your maternal and paternal lineages come from and how strong. they have a very pretty heatmap of the continents for this. National Geographic's DB is probably more sophisticated and ancestry.com also does this. This is pretty cool.
Also included things like curly hair, color of skin eyes and hair.
Like SimaPro says they are described in OMIM. OMIM stands for 'online mendelian inheritance in man' and all the straightforward directly inheritable traits we know of are listed there. its worth taking a look. There are not a lot of known single mutation genetically inheritable conditions.
If the genome sequence includes methylation, then some other traits which are epigenetic, not genetic but would also be determined from a genome sequence. Most famous example of this is MEST which will give an indicator of whether you are a doting parent or not.
Well - a lot if you knew what questions to ask...
The most obvious thing is sex: if you can find the x and/or y chromosomes.
Other things like skin color, eye color, etc, or hereditary diseases and all that would involve running the genome through some sort of database which had values for both pheno and genotypes so you could correlate some of those things and find those physical characteristics.
Scientists are discovering these correlations all the time.
You can go to http://www.ncbi.nlm.nih.gov/omim and search for something like eye color and find out more than you need probably.
The first place to look for answer to this is the 1000 Genomes Project. I feel that their efforts in data analysis offer lessons on what one can and cannot learn from such a dataset as posed in the question.
One obvious thing that is easy to overlook is a whole genome sequence will tell you the mid- to large-sized insertions and deletions in that genome. Any of those that are novel, private or extremely rare (take a word of your choosing) can be used to place that individual in a family tree (if other data are available) and to predict certain phenotypes (from the affected genes). This is also true for some SSRs - simple sequence repeats.
Good haplotype analysis from the genome sequence can predict ethnic ancestry - with a fair degree of accuracy. This is especially so as more genomes are sequenced and available for comparison. Haplotypes of the Y chromosome and mitochondrial genome, give paternal and maternal ancestry, respectively.
If the whole genome sequence gives accurate telomere lengths, then a prediction of age of the individual can be gained +/- 10 yrs.
Disease phenotypes or risk of disease is tricky. For monogenic diseases, this is more accurate and that accuracy falls off as the complexity of contribution to variance of the trait increases. At least a hundred loci contribute to risk of type 2 diabetes, but that risk is also age-dependent and so a young person may not have the disease but only be at increased risk. Furthermore, not everyone with elevated risk will acquire the disease and not all at reduced risk will remain disease-free - for that disease.
Lastly, if this analysis included epigenetic measures, such as methylation of the genomic DNA, then one can predict genome changes from environmental exposure such as tobacco smoking or air pollution. In addition, loci with differential methylation and showing a phenotypic consequence, such as elevated obesity status (BMI) or colorectal cancer risk, are beginning to be defined. This is in its infancy, but the data are growing.
Astonishingly, if they are male you can take a pretty good punt at working out their surname and narrowing down exactly who they are. There was a recent paper on doing this with anonymous genome data, discussed in a blog post here if you want a more accessible account.