Think of an operating system (Linux) as a seed or nut like an almond or walnut. There is the outer part (the applications and system services), and then there is the inner part that contains all the stuff needed for the rest (the kernel). It is generally (with luck) quite small. They are complex and provide all the services needed to deal with hardware, memory, etc.
The kernel is the core of the operating system. In Linux and other Unix and Unix-like operating systems, it is the program that manages all the resources of the computer. In other words, it is the part of the operating system that everything else relys on. It manages all the resources, like memory and devices, which programs are running when, managing and passing signals and interupts, etc.
Linux is the kernel of all Linux-distro operating systems, like Debian, Ubuntu,...
Kernel of an operating system is roughly the part that schedules and dispatches processes - maybe only one, if it's not multiprogramming kernel - and handles the interrupts. Basically it manages the processor itself allocating it to processes and interrupts.
(kernel = core = heart = nucleus = ..., although by 'core' people usually mean the main memory - that used to be made of ferrite cores.)
It's also typical (but not mandatory) that other stuff is added to the kernel, like file systems, some bus devices, memory, and the like.