People talk of telecommunications as 'instant' - but it never is. In fact nothing in life is instant - not even face-to-face conversation. Light and sound move at defined speeds with measurable delays. What matters is the perception of delay - whether it is natural or artificial.
In telecommunications, the technology is rarely the main delaying factor. Human limitations and delays in the system are far more important. So this concept of 'instant' communications is one that has meant different things at different times to different groups of people.
Really, it is all a question of what you are measuring it against...

Originally, sending a telegram was a cumbersome and slow process. You had to send the message using a relatively complex code letter by letter, only to have to reverse the process at the receiving end.
These delays were significantly reduced by teleprinters (in a system known as Telex), which allowed users to type their own messages at a keyboard and send them at the press of a button.
Early telephones had a different kind of delay - calling the operator, giving her the number and waiting while she and possibly other operators involved in the call made the connection entirely manually. This has been speeded up by letting users dial their own calls which are then connected using automatic equipment.
There are sending delays on the Internet as well. It takes time to compose messages, to upload them to the 'server' computer and for them to become available to others. Sometimes the process seems fast - sometimes painfully slow. It all depends on the volume of network traffic - and also what you're used to.

Signal transmission speeds are the fastest thing about telecommunications - but even these are not 'instant'.
Electromagnetic impulses travel at the speed of light - but over distances of hundreds or thousands of miles that means a tiny delay.
A more significant cause of delay is what's being done to the signal to squeeze it down the line. Compression, multiplexing, decoding and decompression - all these steps cause successive minute delays, and there will be more if the signal has to be beamed 40,000 miles out into space, bounced through a satellite and beamed back down to Earth.
The end result of all these delays - a pause of about half a second between speaking on a transatlantic phone call or live television link and your voice reaching the listener - doesn't seem much, but when you're trying to hold a conversation, via space, it can be disruptive and annoying.

As well as delays in sending signals there are also delays in receiving them. It takes time for signals to be decoded and transcribed into ordinary language.
With the telegraph, this meant waiting for the signal to come through in Morse - as audible clicks or in series of needle movements - writing all that down, decoding it, writing it out as a plain language message, giving it to a messenger and actually taking it to the person whose name was on the message.
In the Internet age, the delay is much less - but still measurable. With an e-mail, you have to:
wait for it to reach your domain name server
wait for it to come down the line, through your modem and into your computer
wait for it to be stored on your hard drive
wait to be told that the mail is there
wait to be 'logged on' and able to open it.
Using a high speed connection such as ADSL, the transmission delays are small, but the human and service delays are significant. Usually it takes a few seconds - sometimes it can take hours.

The longest delay to any signal or message is usually the human factor - which depends on how well different methods of communicating fit into the ways in which we instinctively want to communicate.
The delays caused by technology are usually measured in seconds - sometimes minutes. But the delay caused by human factors can take hours, days or weeks.
Take the example of an e-mail. The time from pressing 'Send' to the e-mail reaching the target domain name server is normally less than 30 seconds. Sometimes it can be as much as 15 minutes if the network is very busy. But the delay before the receiving person actually finds that mail, opens it and reads it can be days.
And you might have to wait forever for a reply ...