There are two parts to a game: the theme, and the mechanics. You can start your game design with either one.
The theme is the game's story and setting, if there is one. Some examples are pirates, tending to animals, or piloting a spaceship.
An amusing aside is that non-gamers tend to imagine that a game is only a theme. They share their game ideas with me in the format of a story, and have no concept that such things must actually be represented in game rules and pieces somehow.
You don't even need to choose a theme at this point. Just make individual game objects, and see where they tell you to go.
The mechanics (also called "mechanisms") are the rules of the game. Technically, multiple rules come together to make a mechanic, and multiple mechanics come together to make a game.
Monopoly can be broken down into a handful of mechanics. The "roll-and-move" mechanic is one of them. This mechanic can then be further broken down into three actual rules: You roll the dice, you move your piece that many spaces, and you refer to the space you land on, to see what happens.
Collecting sets of cards, laying tiles, trading, and bluffing, are all mechanics.
If you start your design with a mechanic, the mechanic must be new in some way. It cannot just be "a card-drafting game" or "a war game". If you're a fantastic and successful designer, who can just do the same old thing, but better, and get it published, this doesn't apply to you.
It's difficult to come up with a brand new mechanic, and it's even harder to execute it, to make a great game. I've tried this, and failed, more than once. Almost all games are made up of pre-existing mechanics.
If your design starts with the mechanics, try to create a new combination.
Use existing mechanics, but put them together in a new way. You don't
have to invent a totally new mechanic, yet your game will be
mechanically unique and interesting. How about a word game that involves
an auction? How about a war game which has an element of physical
dexterity?
If you start with the mechanics, you can choose a theme later. However, the theme you choose must, first and foremost, closely match the mechanics of the game.