Heading to Ireland via connecting flight in Madrid

Allowing 1.5 hours to change planes is cutting it close. Will you be booked through on one ticket? If so, if your first flight is late then they're obligated to put you on the next possible flight. If two separate purchases, you'd be out of luck or at least you'd have a lot of convincing to do.

Similarly, if you're booked through on one ticket then they should be able to check your bags through to Ireland.

You should be able to stay within the airport in Madrid so you would not have to clear passport control/immigration. There should be a sign for continuing passengers, something to that effect.

When you make your purchase, the airline websites usually won't let you book two flights that are too close together for a reasonable connection time (again, that's if you're booking the itinerary through on one ticket). Maybe 1.5 hours is what they deem reasonable. Personally I'd try for a longer layover.

I booked through one ticket. Thanks for your help.

If you are all on one ticket, your bags should be checked through to Ireland (make sure when you surrender them in your US airport). You should also get both boarding passes when you check in at your US airport.

On arrival in Madrid, you will have a long walk to the shuttle between T4S and T4. You then have more walking to immigration (passport control). You then go through security again, then more walking to your gate for your next flight. Until you see them, you cannot imagine how HUGE both T4 and T4S are, and you may think I'm emphasizing the walking now, but you won't once you've done it. Seriously, if you have any mobility impairment, arrange now for help through the airport (wheelchair, motorized cart). As you're doing all the walking, you will see signs for how long it will take you. I found the signs accurate - assuming you move at a normal pace and don't stop for anything. There are bathrooms and food vending machines at all gate areas, so don't stop for these until you're at your gate.

On arrival in Ireland, you will pick up your checked bags and go through customs. This usually means just walking through the "Green channel nothing to declare" line. Unless you're "randomly selected for additional screening" (possible but rare), you're done.

I had a two hour connection, and was a bit worried because there was only one flight a day to my destination (Biarritz, France). I made it just fine with some cushion, so 1.5 hours should work (just don't dawdle). From Madrid to Irish destinations, there's likely to be multiple flights a day, so it won't be a problem to be put on a flight later that day if your arriving flight to Madrid is late.

Harold describes the process for connecting through to Schengen countries i.e. France.

Ireland isn’t in the Schengen so follow epitd’s guidance.

Passport control and customs will both occur in Ireland

Oops - I forgot Ireland was not Schengen!

So, the post above is correct, and you shouldn't have to go through security. This means you have more time to get to your next flight. Alas, you'll still have the walking.

There is a train though between terminals right?

Yes, there is an automated train. The signage is very good and you won't get lost - just keep moving!