CPUs are made using small transistors, electrical gates that switch on and off to perform calculations. They take power to do this, and the smaller the transistor is. "7nm" and "10nm" are measurements of the size of these transistors- "nm" being nanometers, a miniscule length-and are a useful metric for judging how powerful a particular CPU is.
For reference, "10nm" is Intel's new manufacturing process, set to debut in Q4 2019, and "7nm" is usually referring to TSMC's process, which is what AMD's new CPUs and Apple's A12X chip are based on.
So Why Are These New Processes So Important?  Moore's Law, on an old observation that the number of transistors on a chip doubles every year while the costs are halved, held for a long time but has been slowing down lately. Back in the late 90s and early 2000s, transistors shrink in size by half every two years, leading to massive improvements on a regular schedule. But further shrinking has come from Intel since 2014. These are the first major shrinks in a long time, especially from Intel, and represent a letter rekindling of Moore's Law.  With Intel lagging, even mobile devices have had a chance to catch up, with Apple's A12X chip being manufactured on TSMC's 7nm process, and Samsung having their own 10nm process. Intel with its AMD's next CPUs on TSMC's 7nm process, Intel's performance in Intel's monopoly on the market-at least until Intel's 10nm "Sunny Cove" chips start hitting shelves.  What The "nm" Really Means
CPUs are made using photolithography, where an image of the CPU is placed on a piece of silicon.
Since smaller transistors are more efficient, they can do more calculations without getting too hot, which is usually the limiting factor for CPU performance. It thus allows for smaller sizes, which reduces costs and can increase the same sizes, and which means more cores per chip. 7nm is effectively twice as dense as the previous 14nm node, which includes companies like AMD to release 64-core server chips, a massive improvement over their previous 32 cores (and Intel's 28).
It's important to note though that while Intel AMD's wants to be twice as fast. Performance does not scale exactly with the transistor size, and at such small scales, these numbers are not as precise anymore. So it's best to take them rather than making exact measurements of power or size. For example, Intel's upcoming 10nm node is expected to compete with TSMC's 7nm node.
Mobile Chips Will See the Biggest Improvements
A node shrink is not about performance; it has huge implications for low-power mobile and laptop chips. With 7nm (compared to 14nm), you could get 25% more performance under the same power, or you could get the same performance for half the power. This means it will be more efficient and much more powerful. We've already seen the A12X chip from Apple crushing some older chips in the marketplace.
A node shrink is always good news, as faster and more efficient chips affect almost every aspect of the tech world. Moore's Law is not quite dead yet. 2019 wants to be an exciting year for tech with these latest nodes, and it's good to see Moore's Law is not quite dead yet.