Pure functions and no side effects
Pure functions are the ones with no side effects. Side effects are bad, because they are unpredictable and make your software hard to test. Let's say you have a method that receives no parameters and returns nothing-this is one of the worst things we could have, because how do you test it? How can you reuse this code? This is not what we call a pure function. What are the possible side effects? Database call, global variables, IO call, and so on. This makes sense, but you cannot have a program with just pure functions, because it won't be practical.
First-class functions and higher-order functions
First-class means that the language treats functions as first-class citizens. In other words, it means having language support to pass functions as arguments to other functions and to return values as functions. First-class function also implies that the language allows you to store functions as variables or any other data structure.
Higher-order functions are related to First-class functions, but they are not the same thing. Higher-order functions often means language support for partial functional application and Currying. Higher-order functions are a mathematical concept where functions operate with other functions.
Partial functions are when you can fix a value (argument) to a particular function, which you may or may not change later on. This is great for function composition.
Currying is a technique to transform a function with multiple parameters in a sequence of functions with each function having a single argument. Scala language does not force currying, however, languages like ML and Haskell almost always use this kind of technique.