I am assuming you are, like almost everyone in North America, using what's called a "Grounded Neutral" supply system in which the Neutral bus at the breaker panel (and at the transformer) is connected to true Ground. In that system you are quite right: Ground and Neutral should have no voltage between them, and Hot should be about 120 VAC compared to either of the other two. Your actual readings certainly indicate a fault in the Ground connection.
When you say Ground, what point are you using? Could be the U-shaped hole in the face of the outlet, of the actual bare bonding wire (Ground) inside the outlet box, or the box metal itself, or some external source.
The way it's supposed to be done is this. Start at the breaker box and the supply cable coming into it from outside. The supply should come into the top of the box with two insulated black lines going to the two input terminals of the main breaker. A third supply line (could be insulated white or just heavy bare copper) goes to the panel's Neutral bus. All the branch circuit white Neutral leads come back to this bus. There is also a separate Ground bus to which all the bare bonding (Ground) lines from the branch circuits are attached. There should be one medium-heavy bare copper Ground lead attached to the Ground bus and exiting the panel to connect to a true reliable Ground source. In many houses this is done with a saddle-shaped ground fitting on the main metal water supply line just where it enters the house, but yours might be different. The Ground bus is directly connected to the panel case itself, establishing its ground. There also is a screw through the Neutral bus that connects it to the panel case, thus ensuring that bus also is at Ground potential.
Now, back out to the outlet at the end of a branch circuit. If all three wires in the cable entering the box are functioning properly, measurements here will be almost identical to those back in the panel. The "almost" there is about the Neutral (White) lead. If there is something using current somewhere in the branch circuit, current will be flowing in both the Hot and Neutral lines, The Neutral wire has real non-zero (although VERY low) resistance, so a current flow through it back to the panel means that, at the outlet, it will have a small voltage on it compared to a true ground. Maybe only a few volts, maybe less than one, but not zero. On the other hand, the bare bonding (Ground) wire should never have a current flowing under normal circumstances (it's only a safe exit route in abnormal conditions), so it ought to have a true zero potential compared to the real "True Ground". That bare wire is supposed to be connected to two points in the outlet box. One is the box itself (via a screw in the back of the box) and the other is the Ground terminal of the outlet device. Now, just for a backup, sort of, the outlet device's Ground terminal normally also is connected to the metal frame of that device, which is in contact with the metal box at the point of mounting with screws. So that's a second means of establishing a Ground connection between the box and the bare wire from the supply cable.
So, suppose everything in the breaker panel is fine. Then suppose the bare bonding lead in the branch circuit between the panel and your outlet box is broken. Although the stuff in the box looks OK, it actually has no Ground - the bare copper lead is just floating unconnected to anything. In that case you'll get the kind of measurements you did. The voltmeter will give you a reading based on microscopic leakage currents (because the voltmeter is a very high-impedance device), but they are meaningless in the cases involving the Ground lead that really isn't. This situation can arise because of a real break in the bare copper wire in the cable, or because somewhere along the branch circuit with multiple outlet boxes, one of the Ground connections is not really connected, and everything after that is ungrounded. On most branch circuits there are several outlet boxes, and in each a cable enters bringing in the supply and it is joined both to the device in the box and to another cable that leaves, carrying the supply on to the next box. To make the junction of bare bonding leads, you are supposed to twist all of them together tightly and then wrap them abound the box's Ground screw at the back and tighten it down. If the exiting cable's bare lead is not connected properly in a box, everything after that box will have no Ground.
Another possibility is that the faulty Ground connection is right at the particular box you are examining. Normally, because the ground is supposed to be connected to both the box back screw and the outlet Ground terminal, and there is a backup connection between outlet frame and box, both of those connections would have to be defective. One way for that to happen is if the bare lead from the cable just is not connected to anything.
Since you have a voltmeter you may be able to track down where the problem is. You already know what you should see in measurements by probing the outlet's slots - you don't even have to open the cover to reach the wire terminals. And you already have recognized measurements that are wrong and indicate a problem. So next you need to establish exactly which outlets (and maybe switch boxes and light sockets) are on the same branch circuit - they all go on and off with the same breaker. Sometimes you can trace out the sequence on the branch, from the panel end all the way to the last box, and sometimes not. Anyway, take your voltage measurements at all the device boxes along the circuit and see if you can identify some that are good and some bad. If there are both good and bad ones, then there is a problem in the bare lead in the last good box, in the cable to the first bad box, or in the first bad box. Shut off the breaker and open the covers in those boxes, unscrew and pull out the outlet device (do not disconnect the wires at its terminals - there should be enough extra wire length in the box to pull it out without disconnection) and inspect the wires carefully with emphasis on the bare leads. If you can, use our meter in resistance measuring mode to check for continuity of each lead from one box to the next. You should find a faulty connection between boxes and maybe be able to see it and fix it.
Now, just suppose the very first box in the branch circuit shows the problem. Now we have two possibilities. The first is similar to what we've just looked at - there is a faulty connection in the bare lead from the panel to the first outlet box. But the other possibility is that the whole system has no Ground connection. See if you can identify a reliable Ground source, like the water supply line through the basement floor. Maybe you don't have that available to you. But if you do, check the voltage between that real Ground and the breaker panel case. It should definitely be really ZERO volts. If it does appear to be zero, do a secpond check -measure the resistance from panel to Ground, and it also should be zero ohms. Anything else on either measurement indicates the case is not Grounded or poorly grounded and your entire household system may be ungrounded. To verify, try the same type of measurements you've already made on other branch circuits. Do you get faulty Ground indications everywhere in the house? That would confirm the lack of a system Ground. Fixing this probably requires a professional electrician.
Be aware of one common way this can happen in older buildings. House wiring systems built up to about 1950, sometimes later, never had any bare bonding (Ground) leads in any of the cables supplying the branch circuits. Every such circuit had Hot and Neutral, but no Ground at all. The best solution to this was to rewire the entire house with new cabling and breaker panel, etc. Another cheaper, but almost as good solution for a few branch circuits, was to install modern Ground Fault Circuit Interrupters (GFCI's) that will do a similar protection function with no Grounds. But some people have tried to solve the problem in a very poor way. They simply replaced the original 2-prong outlets with new 3-prong ones, but since there is no bare Ground lead to use, they just didn't connect what they don't have. So you have an outlet in the wall with three holes in it, but the Ground is totally fake and unconnected! You will get measurements exactly like you have. Maybe that is the actual explanation in your case.