Will the airline wait for me if my incoming flight is delayed (same airline)?

They may wait for you, because they know you're coming, but they won't wait for very long and waiting is not guaranteed. They have a schedule to keep, and there may be operational reasons why they can't hold the flight for long (if it's the last flight of the day, there might be noise restrictions in place after a certain time, or there may be restrictions on air crew working hours).

If you do miss the connection, they will put you up in a hotel for the night.

Do airlines typically wait for passengers who are flying in with the same airline and have a short connection?

Typically not, at least not on US domestic connections. I have personally experienced some egregious "won't wait a single second" scenarios

1. In one case I had a tight connection with a delayed incoming flight but due to no luggage and fast running I made it to the gate with 2 minutes to spare. When I tried to walk on my boarding pass didn't clear. Turned out, they already had given my seat away because, and I quote "we didn't think you would make it".
2. Incoming was late for the last red eye out of LAX. 6-8 passengers on the same flight trying to make the connection. It's tight, but not impossible. Unfortunately we have to go from the far end of Terminal 8 to the far end of Terminal 6. First person gets there maybe one minute late. Plane is still there, jet bridge is still docked, but the gate is already closed. Waiting one more minute would have saved the airline the extra work of dealing with 6-8 stranded overnight passengers and the associated expense.

Typically, if a lot of people are connecting from flight A to B, the airline could wait. In effect, it is better for them to wait maybe 20 minutes than having to provide food and a hotel for loads of people. I have had cases before were airlines waited before.

Alternatively, the airline could fast track you through the airport (e.g. security and passport control if required).

Since there are many answers saying "No", I'll provide anecdotal evidence of a "Yes" answer.

I was flying from Indianapolis to Klamath Falls, OR via MSP and PDX. As I sat in MSP, they announced a delay of the flight to PDX. I politely inquired at the gate to see how long it would be and to check on the likelihood of my making my connection at PDX - they assured me there would be no issue. As I sat there, they announced delay after delay. Each time, I went back to the gate agent, slightly more concerned than the last time, asking about making my connection at PDX, since I knew I was already on the last only flight of the day from PDX to Klamath Falls. They continued to assure me that all would be fine.

On my last trip to the gate agent after the last delay announcement, I was met with, "have a seat, give me a few minutes and I'll get back to you". After being called back up to the gate I was assured that I'd make my connection.

Upon arrival at PDX, I stood up in my seat at the very back of the plane and looked forlornly at the many, many rows of not-moving people in front of me wondering if I'd ever get off this plane, to say nothing of making my next flight. I heard mechanical noise behind me and saw that the rear door had been opened. I heard a flight attendant say "I think it's the guy in the white sweatshirt". Sure enough (after checking), I was wearing the only white sweatshirt around.

A guy in a ground-crew uniform tapped me on the shoulder and asked me to follow him. He led me out the back door, down the stairs, across the tarmac and to the front of the security line for the terminal where my flight was departing. He said, "Hurry, you're the last passenger and they're holding the flight for you. When you get to your destination, contact the luggage people and they'll get an address to forward your checked luggage."

As I boarded the otherwise packed puddle-jumper, the FA welcomed me aboard and closed the door behind me.

So yes, they may well hold the plane for you, and even get you through security (if necessary) in a hurry.

*NOTE: all conversations are approximate - this was 25 years ago on Northwest Airlines (which no longer exists). YMMV.

It varies. They will often wait a little while, especially if it's the last flight of the day and even moreso if there are multiple passengers making the same connection. I've had Delta wait for me after inbound connection delays on several occasions. Actually, I've only had one time where I actually missed a connection on Delta and that was a case where the inbound flight was multiple hours late and I knew I'd miss the connection before I ever departed. They will be more inclined to wait for passengers with higher status levels, but I've had them wait for me even before I had any status with them.

That being said, if the delay is due to weather, ATC flow control, or something else like that which is outside of the airline's control and you do miss the connection, they are not obligated to provide overnight accommodations for you if you miss the connection. They usually won't. However, if the delay is due to crew, mechanical problems, etc., then they will provide a hotel room for you if you miss the connection. In either case, they'll rebook you on the first flight with an available seat the next day.

Personally, especially when connecting to the last flight of the day, I try to book layovers of over an hour when possible, especially at airports and times of year where inclement weather delays are reasonably likely. Of course, this isn't always possible or feasible and sometimes you end up having to book a short connection whether you like it or not.

As a side note, one way that you can help mitigate the risk of missing a connection to a last flight of the day in situations like that is to book the flight on a credit card that provides trip delay protection benefits. For example, Chase Sapphire Preferred, Chase Sapphire Reserve, and (U.S.-issued) Amex Platinum Card* all provide such benefits to one degree or another. These will cover the cost of a hotel if you get stuck overnight due to weather as well as any other reasonable costs you might incur, such as purchasing toiletries that you might need if you're separated from your bag. However, in most cases, these benefits will usually only cover cases where the airline wouldn't cover the expense (such as, for example, if you missed a connection due to weather delays.)

(* Just to avoid a potential point of confusion here, the Amex Platinum Card and the Delta SkyMiles Platinum Card from Amex are two totally different, unrelated card products. The latter does not provide trip delay protection insurance. I have no idea why Amex thought it was a good idea to use "Platinum" and "Gold" in the names of so many completely unrelated card products.)

I fly on Southwest as much as possible for unrelated reasons. There are a number of times they have held the entire flight for two people or even one person: single digit minutes generally. But SouthWest is in general a pleasant airline to choose. That makes it easy to have a positive attitude on it (when you are not the one who is delayed): we're all going to be fine even with a couple of extra minutes added.

Well if the flight youre in is meant to connect all or majority of the passengers it is going to wait ofcourse and make up for that time in flight. But if its a small group then probably not. You will be re-assigned the next flight to your destination

Probably not. Unless a large portion of the passengers on your Kansas to MSP flight are also continuing onto the same destination you are they have very little incentive to hold a flight. It costs an airline roughly $75/minute for a delay.

Not necessarily.

One time I was travelling from Athens to Paris through Frankfurt (with Lufthansa) with estimated waiting time 1 hour. Due to bad weather near Frankfurt, the flight landed in Frankfurt about 40 minutes late. We arrived on our gate literally less than 10 minutes past boarding and we were firmly refused boarding.

After much talking and negotiations (they kept claiming that this was not their fault and there was nothing they could do) they agreed to put us on the next flight to Paris (4 hours later).

Funny thing: In the next flight, the pilot was waiting on the gate for 35 minutes till the catering company bring 2 extra meals for the two extra passengers (us) but the 1st flight did not want to wait for the 2 extra passengers that were delayed in a connecting flight due to weather...

Two complaints to Lufthansa remain to this day unanswered.

Both reservations are on the same ticket, both are with the same airline.

They must either wait for you, for a reasonable amount of time, or reroute. In particular, if your inbound flight is delayed, airline has two options is this case:

1. Delay the outbound flight, along with all other passengers
2. Reroute you, including night accommodation

This because you have a single ticket for a single journey, and the airline is bound to a contract with you. A travel contract. A contract that get you transported from A to B.

It's the airline's decision what the best option is for both. It's the airline's duty to bring you to destination