Well, I'm glad you asked! A "Chelonian" is a reptile belonging to the Order "Testudines". In other words, "Chelonian" is the scientific name for turtles. Except it isn't... It is an outdated scientific name for turtles that used to be used when the Order was called "Chelonia". It is no longer called that, so, technically, the real scientific name of turtles is "Testudine". But... many scientists do still use "Chelonian" as the general scientific name as well. So, umm, really it doesn't matter (Testudine is more widely used though). But does it even matter? Turtles have one of the most confusing (and controversial) taxonomy of any animal, so anything is likely to change... (Like this website... Oh boy...)
A reptile is a little hard to define because of their weird taxonomy, but I will give you some basic traits to help you identify them (their "synapomorphies", if you are a weirdo...). A reptile is usually ectothermic (cold-blooded), this means that their body temperature is the same as the environment they are in. Humans, on the other hand, are endothermic (warm-blooded), which means that we produce our own heat. (Obviously this is a pretty simplified definition, but it does the job...) Reptiles usually have "plates" all around their body. These plates are called scales... Except for when they are not. For example, a turtle has scales, but the scales on its shell, specifically, are called scutes. Easy, right? All reptiles are vertebrates, meaning they have a backbone (the equivalent of a human spine). All reptiles lay eggs... except for those that don't. Yeah, some reptiles, like some certain species of Boa Constrictors, give live birth. Now, if you want to get into the nitty-gritty of what makes up a reptile, I will happily oblige. Reptiles are charaterized by their lack of (or very reduced) tabular bones. These are bones that protrude out of the back of the skull and are not found in any reptiles. Here is an example (the blue parts that the arrows are pointing to):

Now, to get even more scientific. Technically, there are two clades of reptiles: Eureptilia and Parareptilia. Parareptilia contains the older, usually extinct, reptiles. Eureptilia is what you think of as "normal" reptiles: snakes, crocodilians, turtles, lizards, and tuatara (and most dinosaurs and birds... Yeah, don't even get me started on that.). Alright, stay with me, this is about to get really nerdy. So, to distinguish between Eureptilia and Parareptilia, scientists look for holes in the back of the skull (called temporal fenestra). A member from Eureptilia will have, generally, 2 holes there (meaning they are a diapsid), while a member of Parareptilia will not have any holes (meaning they are an anapsid). However, it isn't that easy, some Parareptilians have temporal fenestra, and some Eureptilians lack temporal fenestra. To make it even more confusing, some reptiles have a synapsid skull (where they have 1 temporal opening in the skull)! Although, most synapsids are more closely related to mammals then they are reptiles. (Also, technically some diapsids where the lower hole has closed up are called Neodiapsids or Euryapsids... Basically, because of environmental pressures, one of the 2 holes closed up... At least, that is what we are guessing is what happened... I hope that isn't confusing.) This happens in both Eureptilia and Parareptilia! Here is a diagram hopefully showing the difference between anapsid, synapsid, and diapsid:

This characterization is so weird and fluid that scientists still argue over whether turtles are Eureptilians or Parareptilians, but the general consensus is that they are Eureptilians! So, there you go... A very easy to understand (not) description of what a reptile is!
If you don't know what a turtle is... Never mind. I'm just going to judge you silently... And also tell you (because I'm nice)! A turtle is a reptile (usually belonging to the Order "Testudines", but sometimes not) that has the iconic characteristic of a shell. In extremely simplified terms, a turtle's ribs and spine expanded and fused together to make a bony covering over their body, which some of them can hide in for defense. Some turtles, like Sea Turtles, can't hide in their shells, but they still use it for defense. Technically, if you wanted to trace the ancestry all the way back to the Eunotosaurus (the, as of now, recognized ancestor of all turtles), all turtles (and turtle ancestors, known as stem-turtles) can be found within the Pantestudines clade. Even more technically, all turtles (and stem-turtles) with complete shells are in the Testudinata clade. Also, the Eunotosaurus is in the Pantestudine clade, but not as the common ancestor... That's not confusing at all... Anyways, we will start with the Pantestudine clade to make the stem-turtle activists happy. :)
Let's start this off right:
And here's some fun facts for you guys!