TLDR
The US blocks ships that are coming from Iranian ports
Iran blocks ships that aren't from a select list of friendly countries (we can debate whether they are "blocking" or "not blocking", but the numbers speak for themselves - few ships transited, even before the US blockade).
With some exceptions, like China and India-linked ships, either Iran will want to block a ship or the US will want to block it.
What the US is doing and why
Well, the weird thing about the start of this war was that Iran got increased revenue from its oil:
Once Iran, entirely predictably, started to block Hormuz to "unfriendly countries' ships" (a fuzzy notion that excludes or has excluded at times Russia, China, Pakistan, India and some others, which are "friendly") oil prices started to go up.
Trump at that point wanted to tamper the oil price back down so announced a relaxation of first, Russian oil sanctions and second, and bizarrely Iranian oil sanctions.
Then we get the ceasefire, but Iran puts in place the "toll booth" whereby tankers have to transit Hormuz through Iranian territorial waters and, at least in some cases, pay a $1-2M fee to be let through.
The situation immediately after the start of the cease-fire.
Iran-friendly ships are let through, Iranian oil benefits from a premium and most non-Iran oil is still stuck in the Gulf. This is another unwelcome development. It's a blockade, by Iran, but selective so that it can still make money.
The "solution"?
Add another blockade, American this time, further out, not in Hormuz proper, at the entrance to the Gulf of Oman. Ships "affiliated with Iran" or at least coming from Iranian ports are impeded, by the Americans, not the Iranians. So, in theory, no more Iranian profits.
The problem left over
Ships will not be seized by the US Navy if transiting in Pakistani or Indian territorial waters.
The US will also find it difficult to risk going after Chinese or Indian-flagged ships.
- but they have been pretty good at intercepting "dark fleet", falsely flagged tankers.
Generally speaking, on the US side of things, ships have been either
- left alone (there are widely differing accounts of how many are making it through)
- told to turn back
- herded off to holding areas (more the dark fleet ships than others)
- fired up on rare occasions.
The overall oil supply is even more constrained than previously.
What's going on with Shipping? on April 26th, 2026 offers the qualified opinion that the US's doubled-up blockade is probably working to pressure Iran. While the channel is run by an American who covers shipping matters in depth, he is quite critical of Trump's war planning, so his assessment that it might be working and is not the stupidest idea embarked on so far by Trump in this war shouldn't be dismissed out of hand, I think.
SkyNews YouTube - April 28th, 2026 also claims that Iran may be facing a big problem in 2-3 weeks: running out of storage to put their oil into. Problem being that a wellhead can be damaged, permanently or longterm, if oil flow can't be maintained out (oddly enough, the Gulf States seem at less risk from what others have been saying: "merely" months to get them going again).
What Iran is doing.
Look at any wargame, strategic analysis, movie plot, etc... of the last 40 years and many of them will tell you the same thing: Iran will turn off the world's oil spigot by shutting down the straits of Hormuz. In the 2000s, there was even a bit of a kerfuffle when a US Navy exercise simulating a wargame shutting this type of channel down by an unnamed (but very Iran-like) adversary got gamed: the red team general outsmarted the US by fighting differently and... sunk a bunch of simulated US ships.
(sorry, the wargame was so prescient that I'll quote some of it)
... CJTF-South seized these islands and established a controversial military escort service to protect ships navigating through the region, charging a toll for its services. This led to a direct military response from "Blue," whose primary objectives were to secure international shipping lanes, neutralize Red's weapons of mass effect (WME) capabilities, and restore sovereignty over the disputed islands...
So, when the US attacked Iran on Feb 28, 2026, Iran did exactly what had been predicted. Except that, unlike the 2000s we now have drones and cheap missiles allowing swarming and saturation attacks. Look at the travails of the Russian Black Sea Fleet: this would be a very hard nut to crack.
So Iran can, and did, effectively shut down the strait to anyone it didn't want to transit. That's pretty much everyone except the friendly ones mentioned above. That doesn't hurt the US directly, but it hurts everyone else and it does hurt American consumers when oil prices go up and they need to fill up their SUVs.
And since ships carrying its own oil were, for a while, the only ones allowed through, Iran profited to boot.
p.s. Iran has talked of mines. Whether those have actually been emplaced is unclear. Iran has mines, but any mined area is a problem for everyone, including friendly forces. For now, none of their attacks have seemed to involve mines and no actual mine sightings have taken place. They might use them later, but for now they seem to privilege weapons over which they have more direct control than mines.
p.p.s. There are usually about 140 transits per day in the strait and right now it seems to be running at 5-10 on good days. US military ships have largely stayed out of it except for a few occasions: mines can be a real problem and the area is too narrow for a combat ship to maneuver effectively in.
p.p.p.s.
The effects are pretty big. About 10% of oil supply is shut off. Some portion of Gulf States' oil can be pipelined to the Red Sea (but might be bottled in if the Houthis seriously get involved and blockade the straits off Yemen). LNG supplies Europe relies on are held up. Different countries rely more or less heavily on Gulf-origin oil and some are really hurting, like Sri Lanka. And a big underreported problem is that fertilizers, which often use oil as feedstock, are also effectively blockaded, giving the possibility of crop failures later on.
Here's an image from "What's going on with shipping" that I annotated:

most countries' economies are being damagedyes, however the impact is less in North America which can affect global perception. Also the impact will significantly worsen when the Russia oil at sea is depleted. Russia oil depots in the west are out of service.