Is there a RAW source?
I'll stick to those as best as possible in this answer.
Regarding "What sort of tasks can be performed?":
The main general RAW source on the actual uses of tools is from PHB p.154-157, which describes what sort of tasks can be performed by the various tools and that you get your proficiency bonus when performing those tasks if proficient in the tool. If you haven't read the entries there yet, that's a good place to start for a short list of examples of what tasks can be accomplished by each of the tools (though for Artisan's Tools it only provides a general overview for that category and nothing deeper for each specific tool; the rest have at least some specific examples for each tool).
There's one other general benefit of tool proficiencies from PHB p.187, which is how they affect lifestyle expenses during downtime. Tools that allow you to craft (typically Artisan's Tools of some variety) can earn you up to 5GP a day during your downtime if you have the resources necessary to do the crafting, and grant you a modest lifestyle for free or a comfortable lifestyle at half cost.
To my knowledge, there are no other RAW sources on uses for tools, barring a few references to specifically Thieves' Tools throughout the books and a couple examples from the DMG I refer to regarding ability modifiers for tools checks further on in this answer.
That section of the DMG (p.239) does go into things you can't use tool proficiency for: checks related to the tool (or what the tool produces) that don't actually use the tool. For example, inspecting the quality of woodwork or trying to ascertain its origins doesn't gain the bonus from carpenter's tools proficiency.
Regarding what skill or ability checks can be made with tools, and what general modifiers to use:
To summarize the important rules that are relevant to tools use:
Using tools/performing a tools check is an ability check independent from your other skills. For example, for using Thieves' Tools to lockpick, you do d20 + Dexterity Modifier + Proficiency Bonus (if proficient with Thieves' Tools); none of your other skills (I.E., Sleight of Hand or the like) come into play/have stacking bonuses (and you can only apply your proficiency bonus once to a roll, regardless). All the other tools work the same way, with d20 + ability modifier + proficiency bonus (if proficient with the tool).
Now, if you've read all this in the book and are looking for a definitive listing of what abilities go with what tools, the RAW answer is: There is no definitive ability modifier that each tool uses for its corresponding tool check.
The first paragraph on PHB p.154 speaks to this, saying that tools proficiency is not tied to a single ability as it represents broader knowledge of its use, as well as referencing that woodcarver's tools might use Dex for fine detail work or Str for carving something out of hard wood. DMG p.239 goes further into this, saying that someone using Carpenter's Tools might use Dex for the fine detail on a flute, Int for making a secret door, or Str to build a trebuchet.
As such, by RAW it's left up to the DM to determine what ability modifiers make sense to use with a particular tool use (though you're certainly free to give your input when making the roll if you believe a task is, for example, a more fine-detailed task that would require good Dexterity as opposed to brute muscle if Strength was the original suggested ability).
