What other implementation styles are there besides compilation, interpretation and transpilation?
Fundamentally, there are only the two: compilation and interpretation.
tl;dr: An interpreter runs the program, a compiler translates the program to another language.
An interpreter for language X is a program (or a machine, or just some kind of mechanism in general) that executes any program p written in language X such that it performs the effects and evaluates the results as prescribed by the specification of X. CPUs are usually interpreters for their respective instructions sets, although modern high-performance workstation and server CPUs are actually more complex than that; they may actually have an underlying proprietary private instruction set and either translate (compile) or interpret the externally visible public instruction set.
A compiler from X to Y is a program (or a machine, or just some kind of mechanism in general) that translates any program p from some language X into a semantically equivalent program p′ in some language Y in such a way that the semantics of the program are preserved, i.e. that interpreting p′ with an interpreter for Y will yield the same results and have the same effects as interpreting p with an interpreter for X. (Note that X and Y may be the same language.)
In some cases, we have more specialized names for certain kinds of compilers, depending on what X and Y are, and what the compiler does:

if X is perceived to be assembly language and Y is perceived to be machine language, then we call it an assembler,
if X is perceived to be machine language and Y is perceived to be assembly language, then we call it a disassembler,
if X is perceived to be lower-level than Y, then we call it a decompiler,
if X and Y are the same language, and the resulting program is in some way faster or lighter, then we call it an optimizer,
if X and Y are the same languages, and the resulting program is smaller, then we call it a minifier,
if X and Y are the same languages, and the resulting program is less readable, then we call it an obfuscator,
if X and Y are perceived to be at roughly the same level of abstraction, then we call it a transpiler, and
if X and Y are perceived to be at roughly the same level of abstraction and the resulting program preserves formatting, comments, and programmer intent such that it is possible to maintain the resulting the program in the same fashion as the original program, then we call it a re-engineering tool.

Also, note that older sources may use the terms &quot;translation&quot; and &quot;translator&quot; instead of &quot;compilation&quot; and &quot;compiler&quot;. For example, C talks about &quot;translation units&quot;.
You may also stumble across the term &quot;language processor&quot;. This can mean either a compiler, an interpreter, or both compilers and interpreters depending on the definition.
The terms Ahead-of-Time (AOT) and Just-in-Time (JIT) refer to when compilation takes place: the &quot;time&quot; referred to in those terms is &quot;runtime&quot;, i.e. a JIT compiler compiles the program as it is running, an AOT compiler compiles the program before it is running. Note that this requires that a JIT compiler from language X to language Y must somehow work together with an interpreter for language Y, otherwise there wouldn't be any way to run the program. (So, for example, a JIT compiler which compiles JavaScript to x86 machine code doesn't make sense without an x86 CPU; it compiles the program while it is running, but without the x86 CPU the program wouldn't be running.)
So, we have:

AOT compiler: compiles before running
JIT compiler: compiles while running
interpreter: runs

JIT Compilers
Within the family of JIT compilers, there are still many differences as to when exactly they compile, how often, and at what granularity.
Some JIT compilers for example only compils code once (when it is loaded) and compiles a whole module at a time. Other compilers may gather information while the program is running and recompile code several times as new information becomes available that allows them to better optimize it. Some JIT compilers are even capable of de-optimizing code. Now, you might ask yourself why one would ever want to do that? De-optimizing allows you to perform very aggressive optimizations that might actually be unsafe: if it turns out you were too aggressive you can just back out again, whereas, with a JIT compiler that cannot de-optimize, the program would crash or return a wrong result; in other words, you simply can't allow yourself to perform the aggressive optimizations in the first place.
JIT compilers may either compile some static unit of code in one go (one module, one class, one function, one method, …; these are typically called method-at-a-time JIT, for example) or they may trace the dynamic execution of code to find dynamic traces (typically loops) that they will then compile (these are called tracing JITs).
Combining Interpreters and Compilers
Interpreters and compilers may be combined into a single language execution engine. There are two typical scenarios where this is done.
Combining an AOT compiler from X to Y with an interpreter for Y. Here, typically X is some higher-level language optimized for readability by humans, whereas Y is a compact language (often some kind of bytecode) optimized for interpretability by machines. For example, the CPython Python execution engine has an AOT compiler that compiles Python sourcecode to CPython bytecode and an interpreter that interprets CPython bytecode. Likewise, the YARV Ruby execution engine has an AOT compiler that compiles Ruby sourcecode to YARV bytecode and an interpreter that interprets YARV bytecode. Why would you want to do that? Ruby and Python are both very high-level and somewhat complex languages, so we first compile them into a language that is easier to parse and easier to interpret, and then interpret that language.
The other way to combine an interpreter and a compiler is a mixed-mode execution engine. Here, we &quot;mix&quot; two &quot;modes&quot; of implementing the same language together, i.e. an interpreter for X and a JIT compiler from X to Y. (So, the difference here is that in the above case, we had multiple &quot;stages&quot; with the compiler compiling the program and then feeding the result into the interpreter, here we have the two working side-by-side on the same language.) Code that has been compiled by a compiler tends to run faster than code that is executed by an interpreter, but actually compiling the code first takes time (and particularly, if you want to heavily optimize the code to run really fast, it takes a lot of time). So, to bridge this time where the JIT compiler is busy compiling the code, the interpreter can already start running the code, and once the JIT is finished compiling, we can switch execution over to the compiled code. This means that we get both the best possible performance of the compiled code, but we don't have to wait for the compilation to finish, and our application starts running straight away (although not as fast as could be).
This is actually just the simplest possible application of a mixed-mode execution engine. More interesting possibilities are, for example, to not start compiling right away, but let the interpreter run for a bit, and collect statistics, profiling information, type information, information about the likelihood of which specific conditional branches are taken, which methods are called most often etc. and then feed this dynamic information to the compiler so that it can generate more optimized code. This is also a way to implement the de-optimization I talked about above: if it turns out that you were too aggressive in optimizing, you can throw away (a part of) the code and revert back to interpreting. The HotSpot JVM does this, for example. It contains both an interpreter for JVM bytecode as well as a compiler for JVM bytecode. (In fact, it actually contains two compilers!)
It is also possible and in fact common to combine those two approaches: two phases with the first being an AOT compiler that compiles X to Y and the second phase being a mixed-mode engine that both interprets Y and compiles Y to Z. The Rubinius Ruby execution engine works this way, for example: it has an AOT compiler that compiles Ruby sourcecode to Rubinius bytecode and a mixed-mode engine that first interprets Rubinius bytecode and once it has gathered some information compiles the most often called methods into native machine code.
Note that the role that the interpreter plays in the case of a mixed-mode execution engine, namely providing fast startup, and also potentially collecting information and providing fallback capability may alternatively also be played by a second JIT compiler.
This is how the second generation of Google's V8 ECMAScript engine worked, for example. This generation of V8 never interprets, it always compiles. The first compiler is a very fast, very slim compiler that starts up very quick. The code it produces isn't very fast, though. This compiler also injects profiling code into the code it generates. The other compiler is slower and uses more memory, but produces much faster code, and it can use the profiling information collected by running the code compiled by the first compiler.