The term isotope was coined in 1913 by Margaret Todd, a Scottish doctor, during a conversation with Frederick Soddy. Soddy, a chemist at Glasgow University, explained that it appeared from his investigations as if several elements occupied each position in the periodic table. Todd suggested the Greek term meaning "at the same place" as a suitable name. Soddy adopted the term and went on to win the Nobel Prize for Chemistry in 1921 for his work on radioactive substances.
The concept of isotopes confounded the builders of the Periodic Table in Soddy's time. Things got even worse after J. J. Thompson showed that he could resolve purified neon into neon of two different masses, Ne-20 and Ne-22. It took the birth of quantum mechanics and Chadwick's neutron to put things back together again.
Today we know with confidence that different isotopes of the same element differ in number of neutrons within their atomic nuclei. Neutrons add heft and stability (or instability) to atomic nuclei, without changing the "place" of the element at the table; in other words, what fixes an element's place is the number of protons in its nucleus, not the sum of its protons and neutrons. Thus the concept "at the same place" makes perfect sense for different atomic mass versions of the same element. All naturally occurring elements have isotopes, for example, hydrogen, which has three isotopes so important that they're given quasi-chemical symbols of their own: H, D, and T, corresponding to protium, deuterium, and tritium, having 0, 1, and 2 neutrons respectively.
Our government (and others) have long been in the business of separating isotopes: uranium-235 was the fission fuel for the first atomic bomb, and plutonium-239 was the fission fuel for the second one. The first hydrogen bomb (code-named Ivy Mike) used liquefied deuterium-tritium gas as fusion fuel, i.e., hydrogen molecules consisting of the two heavier isotopes of hydrogen. Ivy Mike weighed around 62 tons, the bulk of which was dedicated to cooling the liquefied fusion fuel. Practical weaponization of the H-bomb was not achieved until lithium deuteride (which doesn't require cryogenics) became the fusion fuel of choice.
Iran is actively pursuing uranium isotope enrichment, ostensibly to collect enough U-235 for either peaceful electrical power generation or for a fission weapon. Less talked about is the concomitant accumulation of so-called depleted uranium (DU) which is the non-radioactive U-238 “waste” obtained during enrichment. DU is both an effective tank armor and a lethal component of bullets or rounds. While travelling at high velocity, DU or DU-coated shells burn into uranium oxide, literally forming a burning projectile. DU weapons and armor were fielded with spectacular results by the US in the First Gulf War: Iraqi tank shells literally bounced off the Abrams tanks equipped with DU armor. You can bet the Iranians were watching that with keen interest.
Isotopes also have many, many peaceful uses: think of radiochemical uses in medicine and biology and their use in determining the geologic age of materials (radiocarbon dating). Stable isotopes like deuterium and carbon-13 also find broad use as detectable labels which can also be introduced into controlled experiments and followed where they go and don't go. Moreover, subtle effects on the rates (speed) of chemical reactions gives insight into how the reactions proceed.
I once worked around neutrons as part of a scientific collaboration. Our endeavors were peaceful, despite occurring in part at Los Alamos National Laboratory. While determining the molecular structure of a certain substance, we needed the help of neutrons to locate hydrogen atoms using a technique called neutron diffraction which uses beams of free neutrons. To make a long story short, we solved the structure, but I went on to show how one could get the same essential information using more conventional instruments, but that’s another story. And that's the closest I ever want to get to loose neutrons.
*My creds include working with neutrons and co-writing a book chapter on isotopes in chemistry.