Sure, I thought I had explained this a few times already. A lease is an option to drill. A contract between the owner of the mineral rights and the oil company.
On federal lands generally the owner of the mineral rights is the federal government. On private lands the owner of the mineral rights may or may not be the surface owner. It’s not unusual in state like TX with along history of oil and gas development for the mineral rights to have been separated from the surface land rights at some point in the past.
So, if an oil company has some interest in drilling an area it may secure a lease or even an option to lease. A least typically pays a “lease bonus” or upfront cash payment and and if a well is drilled a “royalty” which is a percent of the value of the oil and gas produced to the owner of the minerals. You may get a 4 or 5 year lease period sometimes with an option to renew with an additional payment.
Once you have a lease, you may or may not decide to get a permit to drill on the lease. A permit generally requires a specific well plan (e.g., we’re going to drill to this depth and this many lateral feet, etc.) It is generally issued by some state federal agency what ensure the plan is compliant with all the relevant laws. In some cases to drill a well you may need more than one permit (like a land use permit from the locality, and a drilling permit from the state), and of course the pipelines, treating and processing assets you will rely upon need their relevant permits from their relevant agencies too.
Anyway, a permit requires a more specific plan than a lease, and a lease is necessary to get a permit, but even once a well is permitted it is not necessarily drilled. Everything, including the price of the oil and gas and the capital all needs to come together to support the well economics. Sometimes permits are taken in hopes the rest will work out and it doesn’t. Keep in mind a permit is in the thousands or tens of thousand of dollars to obtain and a well may be as much as $10 million to drill and frack. Companies generally pursue leases and permits to keep an inventory on hand but only drill as much as their capital budget (and individual well economic analysis) allows.
But the truth of this should be obvious: any federal, state or local government or regulatory effort to restrict leases and permits and/or the ability to raise capital will all other things being equal result in less domestic supply. The only case where this is not true is when they do something like ban fracking in Vermont where there is essentially no oil anyway. But banning fracking in New York or California or federal lands in New Mexico or Texas clearly and obviously eliminates some prospects that some company otherwise would have leased, permitted and eventually drilled. You don’t get more supply of something by limiting the ability to supply that thing or making it more expensive. This is true of anything, not just oil and gas.