Object Oriented Programming (OOP) is a programming paradigm based on the concept of "Objects".
I'll put it in another way: it's a way of programming where you focus on performing actions on things, or put in terms of language - its about using verbs on nouns.
Let's think about an example: you have an application that is used to track applicants to jobs (an ATS system). The Applicants and Jobs would be the Objects (nouns) in your OOP system. Actions like "apply", "interview", "notifyOffer" "notifyRejection" might be the actions you can perform on these objects.
Simple right? That's because OOP is meant to be a way to model the real world which consists of things like this.
The fundamental concepts are simple, but OOP has a lot of fairly complex academic ideas, design patterns, and principals built up around it. If somebody decides to quiz a candidate on advanced OOP concepts the interviewer can make this quite challenging if they choose to do so.
I don't want to get into the weeds too much in this post but definitely do check out the wikipedia page on OOP to get an idea of the terminology you will hear people throw around.
Functional Programming (FP) is a paradigm where programs are built by creating, composing, and applying functions.
If you recall from basic algebra the concept of a "function" is the same here. This is a thing that takes some argument and returns some result. Remember making graphs where you input "X" and get an output of "Y", and you would call that function f(x) = y ? We are talking about the same thing here.
Functional languages are based on something called Lambda Calculus which is a formal system based on mathematical logic that is used to express computation. It's an underlying foundation of how most functional programming languages are designed.
What does this all mean? Well, one of the advantages of functional programming languages is that in theory you can catch a lot more of the errors in a program at compile time (meaning before it's actually run), because you can mathematically "prove" correctness. In practice though, using strong static type systems in languages like Java allows you to similarly eliminate many types of errors at compile time.
Basically all of them. Or, at least all of the ones you've probably heard of or have seen in job descriptions. Java and C# are the 900 lb gorillas that are both strongly object oriented in their design. Almost every popular programming language you will see advertised has OOP concepts built into it, or at least the ability to program in an object oriented style.
I'll just mention these here so you can get an idea. These are languages where either most everything is an "object" and Object Orientation is a required concept, or they have full support of object oriented programming features.
These are languages that are considered "functional". Programming in these languages is in general a very different experience as compared to the list above.
You can't really call these functional languages but these languages have added support for using many functional programming style paradigms.
There is a funny line on the wikipedia page for Functional Programming:
Functional programming continues to be used in commercial settings.
It's funny, because somebody actually felt the need to mention this. For a very long time in the recent history of programming (probably from about the 90s up until just about literally the past couple of years) Functional Programming had a reputation of being unsuitable for use in commercial projects and strictly limited to academic settings.
This has changed a lot in recent years but in many ways OOP languages still dominate the industry and use of functional programming languages are still in a minority.
Some of this is just momentum: if you teach OOP in schools, and everyone else is using OOP and it's hard in general to simply find programmers (let alone the "right" ones with special skill-sets), you are going to use OOP languages at your company.
The trends are definitely toward using more functional features in all languages and there has been a huge resurgent interest in using functional programming languages for commercial and open source projects, but history has been more in favor of OOP until recently.
The thing you should understand as a recruiter is that there are some programmers that are deeply interested in functional programming. They look specifically for jobs that will allow them to use functional programming languages.
There are also some employers (mostly smaller specialized companies that can afford to be picky) that only use functional languages and are looking for these special programmers that know these languages.
But, if I didn't make this clear already I would be surprised if this group was even 5% of the industry. And part of this is because that a pure functional programming language like Haskell is so different, it's like learning to program in a totally new way for somebody not familiar with this style.
Functional Programming and Object Oriented Programming are different paradigms that programming languages are built upon.
Currently, the industry is dominated by OOP languages, but this trend is changing. Functional languages are becoming more popular, popular OOP languages are getting functional features, and everyone is starting to use these.
There are some candidates and employers where this distinction between programming paradigms is very important.