One of my most vivid memories from school was the day our chemistry teacher let us in on the Big Secret: every chemical reaction is a joining or separating of links between atoms. Which links form or break is completely governed by the energy involved and the number of electrons each atom has. The principle stuck with me long after I'd forgotten the details. There existed a simple reason for all of the strange rules of chemistry, and that reason lived at a lower level of reality. Maybe other things in the world were like that too. (Originally appeared on Facebook's Engineering blog and perfplanet.com.)
A "full-stack programmer" is a generalist, someone who can create a non-trivial application by themselves. People who develop broad skills also tend to develop a good mental model of how different layers of a system behave. This turns out to be especially valuable for performance & optimization work. No one can know everything about everything, but you should be able to visualize what happens up and down the stack as an application does its thing. An application is shaped by the requirements of its data, and performance is shaped by how quickly hardware can throw data around.
Consider this harmless-looking SQL statement:
INSERT INTO some_table VALUES (1, 2, 3);
What is the maximum number of these you can do per second? Well, it first depends on the storage engine you use. If the table is in MyISAM format a write may result in one more disk-seek or rotation than InnoDB. InnoDB stores the row data right after the index. With MyISAM indexes and data are stored in different files. A hard drive can only do one thing at a time, so this detail can make a big difference in performance. It also depends on what mechanisms are in place (write-through caches, journals, etc) to keep random seeks to a minimum, and whether you are using table-level or row-level locking. Digging deeper into how these storage engines work, you can find ways to trade safety for speed.
One way to visualize a system is how its data is shaped and how it flows. Here are a some useful factors to think about:
This model can be applied to a system as a whole or to a particular feature like a search page or home page. It's rare that all of the factors stand out for a particular application; usually it's 2 or 3. A good example is ReCAPTCHA. It generates a random pair of images, presents them to the user, and verifies whether the user spelled the words in the images correctly. The working set of data is small enough to fit in RAM, there is minimal computation, a low mutation rate, low per-user request rate, great locality, but very strict latency requirements. I'm told that ReCAPTCHA's request latency (minus network latency) is less than a millisecond.
How an application is implemented depends on how real computers handle data. A computer really does only two things: read data and write data. Now that CPU cycles are so fast and cheap, performance is a function of how fast it can read or write, and how much data it must move around to accomplish a given task. For historical reasons we draw a line at operations over data on the CPU or in memory and call that "CPU time". Operations that deal with storage or network are lumped under "I/O wait". This is terrible because it doesn't distinguish between a CPU that's doing a lot of work, and a CPU that's waiting for data to be fetched into its cache. A modern server works with five kinds of input/output, each one slower but with more capacity than the next:
The software stack your application runs on is well aware of the memory/disk speed gap, and does its best to juggle things around such that the most-used data stays in RAM. Unfortunately, different layers of the stack can disagree about how best to do that, and often fight each other pointlessly. My advice is to trust the kernel and keep things simple. If you must trust something else, trust the database and tell the kernel to get out of the way.
I'm using approximate powers-of-ten here to make the mental arithmetic easier. The actual numbers are less neat. When dealing with very large or very small numbers it's important to get the number of zeros right quickly, and only then sweat the details. Precise, unwieldy numbers usually don't help in the early stages of analysis. 
Suppose you have ten million (10^7) users, each with 10MB (10^7) bytes of data, and your network uplink can handle 100 megabits (10^7 bytes) per second. How long will it take to copy that data to another location over the internet? Hmm, that would be 10^7 seconds, or about 4 months: not great, but close to reasonable. You could use compression and multiple uplinks to bring the transfer time down to, say, a week. If the approximate answer had been not 4 but 400 months, you'd quickly drop the copy-over-the-internet idea and look for another answer.
So can we use this model to identify the performance gotchas of an application? Let's say we want to build a movies-on-demand service like Netflix or Hulu. Videos are professionally produced and 20 and 200 minutes long. You want to support a library of 100,000 (10^5) films and 10^5 concurrent users. For simplicity's sake we'll consider only the actual watching of movies and disregard browsing the website, video encoding, user comments & ratings, logs analysis, etc.
It's possible to build a single server that holds 10TB of data, but what about throughput? A hundred thousand streams at 300kbps (10^5 * 3 * 10^5) is 30 gigabits per second (3 * 10^10). Let's say that one server can push out 500mbps in the happy case. You'll need at least 60 servers to support 30gbps. That implies about 2,000 concurrent streams per server, which sounds almost reasonable. These guesses may be off by a factor or 2 or 4 but we're in the ballpark.
You could store a copy of the entire 10TB library on each server, but that's kind of expensive. You probably want either:
An unknown factor is the distribution of popularity of your video data. If everyone watches the same 2GB video, you could just load the whole file into the RAM of each video server. On the other extreme, if 100,000 users each view 100,000 different videos, you'd need a lot of independent spindles or SSDs to keep up with the concurrent reads. In practice, your traffic will probably follow some kind of power-law distribution in which the most popular video has X users, the second-most has 0.5X users, the third-most 0.33X users, and so on. On one hand that's good; the bulk of your throughput will be served hot from RAM. On the other hand that's bad, because the rest of the requests will be served from cold storage. A useful hack might be to keep the first minute of every video in RAM at all times.
Whatever architecture you use, it looks as though the performance of movies.example.com will depend almost completely on the number of concurrent reads your storage devices can handle. If I were building this today I would give both SSDs and non-standard data prefetching strategies a serious look.
This subject is way too large for a short writeup to do it justice. But absurd simplifications can be useful as long as you have an understanding of the big picture: an application's requirements are shaped by the data, and implementations are shaped by the hardware's ability to move data. Underneath every simple abstraction is a world of details and cleverness. The purpose of the big fuzzy picture is to point you where to start digging.
 Fortunately there is a newish tool for Linux called "perf counters".
 Jeff Dean of Google deserves a lot of credit for popularizing the "numbers you should know" approach to performance and systems work. As my colleague Keith Adams put it, "The ability to quickly discard bad solutions, without actually building them, is a lot of what good systems programming is. Some of that is instinct, some experience, but a lot of it is algebra."