A battle begins when you bump into another character, or another character bumps into you. There are exceptions. For example, Kagome will not fight Kouga, and Inuyasha will not fight Kikyou etc. Battles also occur when you land on certain squares.
What happens when you get into a battle, and what can you do about it? Below are screen shots of the first screen you see when a battle begins.
If it is your turn (you bumped into another character), then the screen on the left is the one you will see.
Option 1: Fight - Attack the enemy
Option 2: Escape - Run from battle
Text: <OpponentName> has appeared. What will you do?
If it is not your turn (another character bumped into you), then the screen on the right is the right is the one you will see.
Option 1: Defend - Defend against the enemy's attack
Option 2: Avoid - Dodge the enemy's attack
Text: What will you do?
When choosing between Attack or Escape, I would choose attack because most of the time (especially in the later chapters), the opponent's speed is greater than yours, making it near impossible to flee from battle. Also, even if you successfully escape, escaping a battle only gives you 1 experience point, whereas winning a battle will give you more than 1 experience point (usually 2 to 5 experience points).
When choosing between Defend or Avoid, I would choose Avoid because despite how low your speed may be, there will always be a chance to avoid the attack, whereas in defending, if your defence isn't high enough, you have to hope that your opponent misses (which is unlikely).
However, if you fail to avoid the attack, then you will be hit with the full force of the opponent's attack. If you're using an emulator such as VisualBoy Advance which allows you to save/load at any point in the game, this should not be a problem.
2nd Battle Menu
After you select what you want to do (Attack/Escape or Defend/Avoid), the screen should look like one of the pictures above, depending on what option you chose. What all 4 screen have in common are the two options.
Option 1: Confirm
Option 2: Items
Underneath the options, the stats that relate to the option you chose will be displayed. These will usually help you determine if you need to use an item or not.