There are three major properties an edible fat (I am assuming you are not asking about inedible oils like petroleum based products) has that affect how it is best used:
- Smoke point
The flavor of the fat is very important. So called neutral oils (like canola oil or refined grapeseed oil, or refined peanut oil, among many others) have very little flavor and so are suitable for almost any cooking task, as they will not interfere with or compete with the other flavors in the dish.
Other oils such as sesame oil are primarily used for the delicious flavor that they provide, and are almost more of a condiment than an oil for culinary purposes.
In between are oils like olive oil and coconut oil which have a distinctive flavor, but also have good qualities for cooking. Many non-oil fats used in cooking (like butter, or bacon fat, lard, or suet) also have distinctive flavors.
The level of saturation in the fat controls how hard the fat is at room temperature.
Most oils are fairly low in saturation, and are so quite liquid even at room temperature. The major exceptions among the vegetable based lipids are coconut oil and cocoa butter. Coconut oil is solid at room temperature, and cocoa butter is actually quite hard at room temperature.
Note that while you only asked about "oils", many common cooking fats not usually called an oil are much more saturated, including butter, lard, and hydrogenated vegetable oil such as the US brand Crisco.
The level of saturation can affect texture of baked goods made with a given oil, and whether the fat is suitable for the creaming method or in making laminated pastry in baking.
Saturated fats also resist rancidity better over time better than non-saturated fats (although they are still subject to rancidity).
The smoke point of a fat or oil is temperature to which it can be heated before it begins smokng.
Oils which have a high smoke point (such as grapeseed oil or peanut oil among others) are more suitable for high temperature searing, or deep frying.
The oil for a given application is based on the properties it has.
For general cooking uses (by which I am inferring general sauteeing, shallow frying, greasing of casserole dishes, and so on is meant), normally you want a fat which is either:
- Neutral in flavor so as not to interfere with the dish. The neutral oils are all useful for this.
- Complimentary in flavor, to enhance the dish. Many cuisines have traditional fats which are flavorful, but are integrated into their cuisine, such as olive oil or butter.
Saturation matters less (as the fat can be melted before greasing a pan, for example), and smoke point is not terribly relevant unless the application is searing, in which case you want a very high smoke point.
Depending on the application and the cuisine from which a dish derives, almost any oil or fat may be used for general cooking purposes. The major exception is sesame oil, which is so flavorful that it is used as a condiment, not a fat, for culinary purposes.
The most important aspect for deep frying is smoke point. A low smoke point oil would break down and add off flavors to the dish (not to mention making the kitchen smoky and setting off smoke alarms).
Secondarily, for oils that are going to be reused over time, saturated fats such as hydrogenated vegetable shortenings break down more slowly, and may be considered advantageous.
Generally, flavorful oils are not indicated, as the same factors that give them good flavor often give them low smoke points.
Some good oils for deep frying include peanut oil, canola oil, or hydrogenated vegetable shortening.
For most dressings (by which I infer you mean salad dressings as used in many Western style cuisines), the most important factors are flavor and saturation.
You want an oil with good flavor (such as olive oil) or neutral flavor (such as the many neutral oils like canola, refined peanut, corn oil, and so on).
Most dressings also need an oil which is liquid at room temperature, and thus low in saturation. The major exception is the class of hot dressings made with bacon fat.
Many dressings are made from olive oil, or one of the neutral oils.
While you did not ask about baking applications, there are two types of baking which require a fat with specific properties: cakes made by the creaming method, and laminated pastries like croissants or puff pastry.
These applications require a fat which is solid at room temperature, but still plastic enough to be manipulated. Generally, only butter and hydrogenated vegetable oil are suitable.
The oil or fat you choose for any particular purpose will be based on what is best for that application—but a good neutral oil with a high smoke point such as refined peanut oil or canola oil can serve the vast majority of culinary purposes most of the time.