Thinking about atoms in biology
This is not an introduction to chemistry or even to atomic theories. It's an introduction to what atoms are with the aim of providing enough background to be useful for understanding molecular biology.
Introduction
"Nothing exists except atoms and empty space; everything else is opinion." (Democritus c. 400 BCE)
Atoms are the building blocks of the world. There are different atoms, which differ in their weights and the way they can interact with each other. So for example lead atoms are heavier than oxygen atoms. Different atoms can join with each other. For example, hydrogen atoms can attach to oxygen atoms to make dihyrodgen oxide - H2O.
Biologists don't normally need to think about individual atomic structures, and we don't need to know what they atoms made of. Instead, biologists normally think about atoms which join with each: molecules. In the example above three atoms (two hydrogens and one oxygen) combined to form a single molecule (H2O).
Biological chemicals you've heard of are molecules: proteins, vitamins, fats and cholesterols, DNA etc. These are all molecules that have been classified into types based on something peculiar to them. So proteins are made by cells and composed of amino acids (see below), but by contrast, vitamins have nothing in common on an atomic level and are classified as vitamins due to their dietary role.
It's worth getting some perspective on the scale we're talking about here. DNA is a huge molecule, and back of the envelope calculations ([link]) says that the amount of atoms in the DNA in a single human cell is 204 billion. That's just in one cell, and the human body is made of trillion of cells, and those cells are composed of heaps and heaps of molecules, not just DNA.
From atoms to molecules
Atoms that connect to each other form molecules. Most of the time, biologists don't need to know much about the molecules they work with. They just need to know what they do. It might be important to know whether a molecule is big or small, and whether it attaches itself to water or fats, and any other special properties it has.
An important type of molecule are proteins. Proteins are big molecules built out of small module molecules called amino acids. The important thing to know about amino acids is that because they're ubiquitous biologists are often quite familiar with them and their properties. But even though biologists are familiar with amino acids, most of the time it's not important to know what amino acids a protein is actually made out of. Instead it's important to know how the protein behaves. In cases when it's not known how the protein behaves then it might be helpful to know how it is expected to behave, based on its shape and the amino acids it's made of.
From atoms to proteins
You might know that two different proteins, for example Erk and Mek, interact with each other. The names of proteins often don't mean very much. What's more important is to think about Erk and Mek as two unique shapes and that their shapes are determined by their amino acids. You might also think about what actually happens when Erk and Mek interact, whether they exchange any atoms or whether they force each other to change shape.
Molecules are sometimes atoms connected in chains, sometimes as webs, and most of the time as a mixture of the two: long lines of linked atoms with other atoms coming off in various directions. But that's just their structure on paper. Their shape is far more complicated. Molecules can be envisioned as atoms (small balls) connected to each other by forces (sticks). But the connections aren't rigid and so molecules often fold in on themselves or wrap around themselves to form complicated shapes.
Amino acids are small molecules that connect to each other in a line. But once that line is made they squiggle around themselves to form a unique structure that is very important to their function. Their structure determines which atoms are facing outwards and can touch other molecules, and also determines their shape, allowing them to do things like latch onto other proteins.
Final notes
Atoms like hydrogen, carbon, nitrogen, oxygen, and others are important building blocks of the world. When atoms join one another they form molecules, and there is a special list of molecules called amino acids that join together to make up proteins.
What I've written above is sufficient for a basic biological understanding. But of course the more you know the better. A background in organic chemistry might help you predict what a change in conditions (e.g. pH or redox potential) might have or how different molecules may interact. A familiarity with amino acid chemistry helps understand the shapes and activities of proteins. Some amino acids are known to act as disruptors of beta-sheets while other offer potential for forming reducible bonds. And of course, not everything is organic chemistry, and an understanding of ions and inorganic compounds can be valuable.
Caveat: References: Myself and some google.