To expand on the first answer about POSIX, to understand what "unix-like" means, well first one should try to understand what exactly is UNIX. Looking at the documentation from the Open Group, which owns the Unix trademark, you'll find details about the Single UNIX specification's evolution - here's UNIX03:
The UNIX 03 Product Standard is the mark for systems conforming to
Version 3 of the Single UNIX Specification. It is a significantly
enhanced version of the UNIX 98 Product Standard. The mandatory
enhancements include alignment with ISO/IEC 9989:1999 C Programming
Language, IEEE Std 1003.1-2001 and ISO/IEC 9945:2002. This Product
Standard includes the following mandatory Product Standards:
Internationalized System Calls and Libraries Extended V3,Commands and
Utilities V4, C Language V2, and Internationalized Terminal
The UNIX 98 Product Standard is a significantly enhanced version of
the UNIX 95 Product Standard. The mandatory enhancements include (1)
Threads interfaces, (2) Multibyte Support Extension (MSE), (3) Large
File Support, (4) Dynamic Linking, (5) changes to remove hardware
data-length dependencies or restrictions, and (6) Year 2000 changes.
In addition the following optional enhancements are included: Software
Administration facilities and a set of APIs for realtime support. This
Product Standard includes the following mandatory Product Standards:
Internationalized System Calls and Libraries Extended V2,Commands and
Utilities V3, C Language, Transport Service (XTI) V2, Sockets V2 and
Internationalized Terminal Interfaces. In addition, it may also
conform to the Software Administration Product Standard.
This Product Standard defines a consolidated platform for the support
of a wide range of applications originally developed for one of the
class of operating systems that were derived from the UNIX Operating
System code and/or interfaces originally developed by AT&T, in
addition to the facilities provided by the Base Product Standard. It
has wider scope than Base. This Product Standard includes the
following Product Standards: Internationalized System Calls and
Libraries Extended, Commands and Utilities V2, C Language, Transport
Service (XTI), Sockets and Internationalized Terminal Interfaces.
Server versions of the standard add Internet Server and IPv6 in some cases.
So of course we see the reference to AT&T Bell Laboratories and the C language is at the heart of what UNIX is: the C language, modular base tools, and the shell and how the kernel, filesystem and other key OS components were designed and implemented.
That's where the book The Design of the UNIX Operating System by Maurice J. Bach becomes invaluable reading because it's historical matters at this point. Of note of course is how this is related to other inventions like the C language indeed. C was developed by AT&T Bell to implement Unix with a language that could be as fast as assembly but portable across different hardware, and a lot of POSIX is an extension to standard C.
Insofar as the kernel itself is concerned, you'll often find a conceptual diagram such as this one to illustrate what a UNIX kernel traditionally was about:
Here are some excerpts from Mr Bach's classic book (1986) which discuss the foundations of the UNIX System V kernel:
However, they [application subsystems and programs] all use
lower-level services ultimately provided by the kernel, and they avail
themselves of those services via the set of system calls. There are
about 64 systems calls in System V, of which fewer than 32 are used
frequently. They have simple options that make them easy to use but
provide the user with a lot of power. The set of system calls and the
internal algorithms that implement them form the body of the
[...] its two major components are the file subsystem and the process
Files are organized into file systems, which are treated as logical
devices; a physical device such as a disk can contain several logical
devices (file systems). Each file system has a superblock that
describes the structure and contents of the file system, and each file
in a file system is described by an inode that gives the attributes of
the file. System calls that manipulate files do so via inodes. [and
the buffer pool]
[...] There are two versions of the inode: the disk copy that stores
the inode information when the file is not in use and the in-core copy
that records information about active files.
The execution of user processes on UNIX systems is divided into two
levels: user and kernel. When a process executes a system call, the
execution mode of the process changes from user mode to kernel mode:
the operating system executes and attempts to service the user
[...] the philosophy of the UNIX system is to provide operating system
primitives that enable users to write small, modular programs that can
be used as building blocks to build more complex programs. One such
primitive visible to shell users is the capability to redirect I/O.
[...] In addition to servicing system calls, the kernel does general
bookkeeping for the user community, controlling process scheduling,
managing the storage and protection of processes in main memory,
fielding interrupts, managing files and devices and taking care of
system error conditions.
If you're interested with the different implementations of kernels in unix-like operating systems, you can also take a look the the FreeBSD implementation(4.4BSD) or at the Mach kernel or look at this comparison of their features.
The more you know about the design of UNIX, the more you understand what happened in the following diagram about the ancestry of UNIX and its history. Mr Bach is talking mostly about System V in his book but he discusses BSD too:
There's more to this than meets the eyes really. For instance, Mac OSX is UNIX03 certified but do you see it connected to any of the pure UNIXes (in red mostly)?
Above you can see how BSD, GNU, Microsoft and diverse individuals contributed to this universe. Even though GNU and ultimately linux have no direct lineage to UNIX, you see that GNU is an effort to re-engineer in the open source world the tools and software from commercial UNIX that had become closed. So looking at the GNU maintained software gives an idea for instance at the initial prototype apps and libraries.
Licensing wars played a role in the evolution(and stagnation sometimes) of UNIX. You can see immediately that UNIXes are lined up according to license type - closed vs. BSD (BSD allows for making the code closed source... see OSX) and GPL which allows Linux and GNU to complement themselves in the copyleft world. Here's the classic map of the linux kernel initially developed by Linus Torvalds, which also reveals what a kernel "can" be in a Unix-like operating system:
This hints at the idea that a "kernel" design type is not what makes the UNIX standard or what defines a unix-like OS. This is evidenced by the fact that many unix-like OSes may have either a monolithic kernel or a microkernel - monolithic was the classical design type for UNIX. In fact, even within pure UNIXes, HPUX has a monolithic kernel whereas AIX uses a microkernel. This debate about design is about performance and is not related to Unix ancestry or identity. On the other hand, there is a traditional conceptual approach to providing services to software, dealing with file systems etc. under UNIX/unix-like operating systems.
I believe such considerations will add context to the OS part of your question.