Sounding line
Any piece of rope with a weight will do in a pinch.
A sounding line is, as simple as the name suggests. A line. This line is used to 'sound'. Which has nothing to do with actual sound, like the vibrations in the air you can hear with your ears. No it is a tool used to measure the depth of the water. The line is weighted at the end, and marked to measure distance.
By dropping the line overboard the side of a ship it can be used to measure the depth. The whole line is not thrown in with a wide trow, no, the line is lowered and kept taught by the person who operates it. The moment the weight hits the bottom, the operator feels the line go slack and knows they have reached the bottom. Now the markings on the line are checked and the water depth is known.
A good operator can do this on a moving ship, trowing the line in, just enough forward to make it be vertical when the line goes slack. A general knowledge of the expected depth is required of course. But that can also be obtained in two or three trows.
UKC
One must remark that the measurement is from the waterline to the seafloor. The draught of the ship should be subtracted from that value to know the Under Keel Clearance. Although many ships can sail with a very minimum UKC, they cannot go fast, or manoeuvre very well when they are close to the bottom, as the presence of the hull that close to the bottom creates a narrow opening for water to flow trough, sucking ship down.
No sounding on this line
The line to be practical has a limited length of course. So if the sea is of sufficient depth, "no sounding on this line" is called out to signal that it's beyond the reach of the line that is being used. So deep enough, or a longer, more cumbersome, line has to be used.
Sounding staff
For smaller boats, or shallower waters also a long stick can be used, aptly named 'sounding staff', which is faster to use and more accurate at being used faster.
Sea floor information
The bottom side of the weight on the end of the line, is often made slightly hollow, or concave, and can be smeared with a sticky substance, this will pick up anything that is on the bottom. And this way the composition of the bottom of the sea can be found out. Can be rocks, pebbels, sand, mud, shells. This can determin on how close a captain wishes to sail their ship to the bottom. Mud will mean there's a very minimal danger of damaging the hull, but a sharp rocks will rip a hole trough even the strongest hull given the change.
Modern alternatives
Modern ships don't use a rope anymore, but they use an electronic device, that uses echo location technology. A speaker sends out a 'ping' and a microphone picks that signal back up. The time it took for this ping to travel trough the water bounce on the bottom and be heard again by the microphone, can be used to calculate the depth. As the speed of sound trough water is a known value.
But there can be occasions where you don't want to use an Echo Sounder, you might not want to be heard by others who are listening. Or you don't have one, or it is broken.
Even this modern alternatives don't have infinite range. Most regular ships their echo sounder only works up to fifty meters of depth. There is no point in knowing the exact depth beyond that. Normal practise is to turn the device off when deep water is reached, as not to disrupt the wildlife that may be effected from it.
Then why turn it on in shallower waters? Because running a ship aground and possibly spilling oil is a much bigger problem for the fish.
Do they still carry a manual version onboard (which is not just a rope and a stone at the end) in case the electronic version breaks?
Yes! Although it's purpose is to sound around the ship, every few meters, when a ship has run aground. So one can map where the ship is stuck, how much of the ship is in the mud/sand/rock. If the ship has been pushed up. All things you need to know before a tug will try to pull you off. As it determents how to pull, and with how much force.