Many statistical jobs ask for experience with large scale data. What are the sorts of statistical and computational skills that would be need for working with large data sets. For example, how about building regression models given a data set with 10 million samples?
Good answers have already appeared.  I will therefore just share some thoughts based on personal experience: adapt the relevant ones to your own situation as needed.

For background and context--so you can account for any personal biases that might creep in to this message--much of my work has been in helping people make important decisions based on relatively small datasets.  They are small because the data can be expensive to collect (10K dollars for the first sample of a groundwater monitoring well, for instance, or several thousand dollars for analyses of unusual chemicals).  I'm used to getting as much as possible out of any data that are available, to exploring them to death, and to inventing new methods to analyze them if necessary.  However, in the last few years I have been engaged to work on some fairly large databases, such as one of socioeconomic and engineering data covering the entire US at the Census block level (8.5 million records, 300 fields) and various large GIS databases (which nowadays can run from gigabytes to hundreds of gigabytes in size).

With very large datasets one's entire approach and mindset change.  There are now too much data to analyze.  Some of the immediate (and, in retrospect) obvious implications (with emphasis on regression modeling) include


Any analysis you think about doing can take a lot of time and computation.  You will need to develop methods of subsampling and working on partial datasets so you can plan your workflow when computing with the entire dataset.  (Subsampling can be complicated, because you need a representative subset of the data that is as rich as the entire dataset.  And don't forget about cross-validating your models with the held-out data.)


Because of this, you will spend more time documenting what you do and scripting everything (so that it can be repeated).  
As @dsimcha has just noted, good programming skills are useful.  Actually, you don't need much in the way of experience with programming environments, but you need a willingness to program, the ability to recognize when programming will help (at just about every step, really) and a good understanding of basic elements of computer science, such as design of appropriate data structures and how to analyze computational complexity of algorithms.  That's useful for knowing in advance whether code you plan to write will scale up to the full dataset.
Some datasets are large because they have many variables (thousands or tens of thousands, all of them different).  Expect to spend a great deal of time just summarizing and understanding the data.  A codebook or data dictionary, and other forms of metadata, become essential.

Much of your time is spent simply moving data around and reformatting them.  You need skills with processing large databases and skills with summarizing and graphing large amounts of data.  (Tufte's Small Multiple comes to the fore here.)
Some of your favorite software tools will fail.  Forget spreadsheets, for instance.  A lot of open source and academic software will just not be up to handling large datasets: the processing will take forever or the software will crash.  Expect this and make sure you have multiple ways to accomplish your key tasks.
Almost any statistical test you run will be so powerful that it's almost sure to identify a "significant" effect.  You have to focus much more on statistical importance, such as effect size, rather than significance.
Similarly, model selection is troublesome because almost any variable and any interaction you might contemplate is going to look significant.  You have to focus more on the meaningfulness of the variables you choose to analyze.
There will be more than enough information to identify appropriate nonlinear transformations of the variables.  Know how to do this.
You will have enough data to detect nonlinear relationships, changes in trends, nonstationarity, heteroscedasticity, etc.  
You will never be finished.  There are so much data you could study them forever.  It's important, therefore, to establish your analytical objectives at the outset and constantly keep them in mind.


I'll end with a short anecdote which illustrates one unexpected difference between regression modeling with a large dataset compared to a smaller one.  At the end of that project with the Census data, a regression model I had developed needed to be implemented in the client's computing system, which meant writing SQL code in a relational database.  This is a routine step but the code generated by the database programmers involved thousands of lines of SQL.  This made it almost impossible to guarantee it was bug free--although we could detect the bugs (it gave different results on test data), finding them was another matter.  (All you need is one typographical error in a coefficient...)  Part of the solution was to write a program that generated the SQL commands directly from the model estimates.  This assured that what came out of the statistics package was exactly what went into the RDBMS.  As a bonus, a few hours spent on writing this script replaced possibly several weeks of SQL coding and testing.  This is a small part of what it means for the statistician to be able to communicate their results.
