When we say "chemistry", we are almost always talking about things that happen in the electron cloud of atoms. Electrons are gained, lost, shared, redistributed, and polarized to give rise to the richness of chemistry. One of the things that we hold constant is the nucleus; in fact, this is the basis of how we balance chemical equations... every nucleus that goes in must come out. The reason we can do this is that, when compared to how electrons react, nuclei are "constant", but there are reactions that take place in the nucleus and between nuclei! When we look at the energetics of electron chemistry compared to nuclear chemistry, we find that the energy of these reactions is so significantly different that we can (usually) treat them independently... nuclear reactions are not affected by the electrons in an atom, and vice versa (to some extent).
Particle Types:
Think of "matter" kind of like a forest. From a distance, it looks pretty solid, but when you get up close you see that it's a lot of tree trunks with a lot more empty space between them. An alpha particle is like a big truck trying to drive through the forest... it will do a lot of damage to the first couple trees it hits, but it will be stopped before it gets too far into the forest. A beta particle is like a bullet being fired into the forest... it can get between a lot of the trees, but eventually it will probably hit one of the trunks and damage that tree, perhaps pretty severely. A gamma ray is like a high-powered laser being fired into the forest... it will penetrate deeply and might even pass through some of the thinner trees or branches, doing a lot of damage along the way before it is blocked by a bigger tree.
Fission or Fusion
Balancing nuclear reactions - these actually follow the same rules as every other chemical reaction, we just have to adjust our counting a little bit. When we balance a nuclear reaction, the identity of the elements going in and coming out can change, but the total number of protons and neutrons must be the same. We're still using the idea that "matter cannot be created or destroyed", but now we're recognizing that nuclei can break or fuse.
Nuclear reactions often occur as a series of individual steps until (relatively) stable nuclei are formed.
Nuclear decay follows 1st-order reaction kinetics which means that its half-life is not dependent upon the amount of substance that is present. Half-life is the amount of time it takes for half of a substance to react. For reactions that do not follow 1st-order kinetics, the amount of time required for half of the original substance to react is not the same as the amount of time required for half of the remaining half to react.
This is one of the reasons radioactive substances are so potentially useful and potentially dangerous. If I start with 100.0g of a radioactive substance and wait for 1 half-life to pass, I will have 50.0g of radioactive substance remaining. If I wait for a second half-life to pass, I will have 25.000g of radioactive substance remaining. After a third half-life, 12.5g of radioactive substance will remain. That's why it takes so long for some radioactive substances to decay to undetectable or background levels of radiation.
The half-life equation in the book:
{amount remaining} = {initial amount} x (½)n
Can be rearranged a couple ways. First, we can just rearrange to get to the fraction remaining:
{amount remaining} / {initial amount} = {fraction remaining}
{fraction remaining}= (½)n
If you multiply the "fraction" by 100, you get percent remaining. We can solve that expression for "n" by taking the log of both sides and rearranging a little bit...
log{fraction remaining}= n log(½)
n = log{fraction remaining}- log(½)
Most half-life problems are some rearrangement of one of these expressions.This is a great example of "units matter". Radiation has a number of properties and effects, so there are a number of different ways to report "how much" radiation using different units.
Bq or Ci - activity or decay rate - this describes how many decays happen over a given time span. This unit does not describe what type of nuclear reaction is happening or how dangerous or safe a given sample is. These units can be used to quantify how much radioactive substance is presence .
rad - tissue damage - rad is a measure of how much thermal energy is being absorbed and potentially causing damage to tissue. This is a measure of the total amount of radiation without a specific time period. It's like "how much water did you drink?"... If I drink 4 gallons of water over the course of a week, it's not a big deal, but if I drink 4 gallons of water in 2 hours there will be a big problem. A rad does not consider radiation type or the type of tissue being exposed.
rem - tissue damage - rem is just a modified version of rad where the type of radiation and tissue are incorporated. An alpha particle striking the bottom of my foot is probably not the same as a gamma ray striking my brain.
Rems (and rads) are ways to measure a cumulative radiation exposure. We are all exposed to radiation every single day from both natural and non-natural sources. It's not a hazard that can be eliminated, but we can manage our exposure.
Radioactive substances can be used to solve a variety of problems and answer many questions for three principle reasons:
Nuclear medicine is a constantly evolving field and has provided some amazing breakthroughs in diagnostics and disease treatment. If you're interested in this field, do a web search on "nuclear medicine" and you'll find plenty of information. Here's a brief summary of a report from the National Academies Press: https://www.nap.edu/resource/11985/advancing_nuclear_medicine.pdf
Most of the large-scale power generation technologies use the same (very simplified) process: some heat source is used to boil water, the steam from the boiling water is used to spin a turbine, the spinning turbine generates electricity. Burning fuels (coal, oil, natural gas, ethanol, biomass, etc) are a source of heat. Geothermal vents are a source of heat. Nuclear reactions are a source of heat. Focused rays from the sun are a source of heat.
All of these technologies (and others) have hazards. Procuring the fuels (nuclear or otherwise) imparts environmental harm and is hazardous to the people who are involved. Building and maintaining the various technologies have hazards and costs. Waste must be managed, whether it is waste from fuel-based technologies (ash, smokestack emissions, etc), nuclear technologies (spent radioactive waste), or just the waste generated when equipment wears out.
Nuclear power generation is not inherently "better" or "worse" than any other technology, it just has different hazards.