To understand how memory works, lets start with the functions of different parts of the brain:

The prefrontal cortex processes short term memory, such as a sign on the highway or the room number of a hotel room.
The amygdala helps store conscious and unconscious emotions. It also stores emotional memories.
The hippocampus is used for transferring memories from short term to long term.

Now what is the difference between short term and long term memory anyway?

Short term memory lasts about 15 to 30 seconds and is supported by regular brain activity in the prefrontal cortex.
Long term memory actually has a physical presence in the brain, stored in the prefrontal cortex so that it can be accessed later.

There are two categories of long term memory:
Implicit– remembered habits and skills
Explicit- things that we are actively trying to remember. There are two types of explicit memories: episodic memories (things that happened to you) and semantic memories (general knowledge)

How does the brain actually store these long term memories?

a process called encoding. The hippocampus links all relevant information together into one memory by connecting neurons so that signals can pass from one to the other. Basically, it’ll connect elements of one event into single episode or memory so you don’t have separate memories of how that event smelled, looked etc. It also decides what information is important to keep in the long term memory. Priorities include memories with strong emotional components and memories rehearsed repeatedly in short term memory (which is how you retain the information you study, for example).

Once a new memory is formed, the neurons with that memory stay in the hippocampus for a while before moving further in the cortex. Long term memories are stored based on the strong component of that memory–for example, a spoken memory will probably be stored near the language centers. You can also have memories stored in multiple places.

Why are some memories stronger than others?

Whenever you think about a memory, you make that memory stronger in your mind. By reactivating the memory, you can cause the memory to be stored again in different part of your brain, literally increasing it’s “rent space” in your brain. Due to this factor, memories are always being changed and updated, so you won’t remember your memories as you did the day before (if you’re thinking about them).

If our memories changes, how can we be sure that we remember them correctly?

The answer is that we can’t be sure. Because our memories can evolve as we think about them, it’s very easy for them to change based on our thoughts. Studies have shown that many innocent people confess to crimes they did not commit because of a false memory that was implanted in their head. These people have no reason to confess–in fact, they have much more reason to lie; and yet, they are still convinced they committed a crime. As our memories are subject to change, they are by nature unreliable.