Every single command and method call you make has to be executed by anyone thread. From that thread's perspective, it is itself the current thread. So in other words: Thread.sleep(delay) pauses the thread that executes the Thread.sleep() method.
Also, keep in mind that multi-threading and multiple cores only have a very distant relationship.
Even before multi-core CPUs were commonplace, pretty much every operating system supported heavy multi-threading (or multi-tasking, which is basically the same thing for the purpose of this discussion) operation.
In modern OS this is done with a technique called preemptive multitasking. This basically means that the OS can forcibly pause the currently running process and allow another one to run for a short time, providing the illusion of actual parallel processing.
And since a lot of time in a given process is often spent waiting for some external I/O (network, disk, ...) that even means that you can use the CPU more efficiently (since the time a process would spend waiting for IO another process can spend doing actual computation).
As an example at the time of writing this, my laptop has 1311 threads (most of which are probably sleeping and only a handful will actually run and/or wait to run), even though it has only 4 cores.
tl;dr while multiple cores allow more than one thread to actually execute at the exact same time, you can have multi-threading even with a single core and there's very little noticeable difference if you do (besides raw performance, obviously)