A natural resource is any useful resource to humans that is produced by natural processes. There are two major kinds, renewable and non-renewable. Renewable resources are those that are constantly being re-supplied by nature, like sunlight and wind. We cannot use them faster then they are being supplied. Non-renewable resources are what you are talking about. They are resources that humans are using up faster than nature can replenish the supply. Crude oil is a perfect example. We are extracting and using crude oil from the Earth's crust much faster than it can be made by nature (it takes millions of years). Ultimately, the danger is that we will use up all of the crude oil and will be faced with a situation where we are forced to cope without it in a rapid timetable. But things get difficult well before we reach that point.
There is a concept out there called "peak oil". This means that there will come a time where there is still plenty of oil left in the ground but all of the readily accessed oil has been extracted so we are forced to extract the more difficult oil supplies. This means that extracting oil will be more difficult, slower, and much more expensive than we are accustomed to. This in turn lowers the effective supply of oil, thus increasing the demand. So while there is still oil left to extract and utilize, the price increases dramatically and the supply gets smaller and smaller over time. So we start feeling the effects of depleting a non-renewable resource long before it actually becomes depleted. This is why it is easier and less painful to start weaning ourselves off of current levels of oil consumption now while there is still time before the effects of peak oil take hold and make the transition more immediate and difficult.