What does this syntax mean?

$ type echo
echo is a shell builtin

meaning, the echo command is part of the bash program itself (assuming you use bash)

-n is an option, so let's see what it does

$ help echo
Write arguments to the standard output
...
-n  do not append a newline

So when we run the line:

zanna@monster:~$ echo -n "Today's date is: "
Today's date is: zanna@monster:~$ 

Hmm that doesn't look very good, because there's no newline after the printed text. We'll come back to this.

$ type date
date is /bin/date

ah so the date command is a separate program. What does it do?

$ man date
Display the current time in the given FORMAT, or set the system date.

The characters after the date command are format options (which must be preceded by +) - different parts of the date are specified (for example %A is the full name of the day of the week - see the rest of man date for the complete list of options)

$ date +"%A, %B %-d, %Y"
Tuesday, February 7, 2017

So if we put the commands together in a script and then run the script we will get

Today's date is: Tuesday, February 7, 2017

Nice! If you want the same effect in a terminal, you can use a semicolon to separate the two commands instead of a newline:

$ echo -n "Today's date is: " ; date +"%A, %B %-d, %Y"
Today's date is: Tuesday, February 7, 2017

You should start with manual pages, the command man. Just type man <command> to get information about a <command>. Navigating in man is not very intuitive but there are lots of guides to it, for example https://wiki.gentoo.org/wiki/Man_page/Navigate#Navigating_and_searching_man_pages.

Relevant parts of man echo and man date:

echo

echo [SHORT-OPTION]... [STRING]...

Echo the STRING(s) to standard output.

-n     do not output the trailing newline

So it prints the string and does not go to the new line after that (which is the default behavior), so the output of the next command will be printed on the right side of the echoed string.

date

date [OPTION]... [+FORMAT]

FORMAT controls the output.  Interpreted sequences are:

%A     locale's full weekday name (e.g., Sunday)

%B     locale's full month name (e.g., January)

%d     day of month (e.g., 01)

%Y     year

By default, date pads numeric fields with zeroes.  The following optional flags may follow '%':

-      (hyphen) do not pad the field

I hope it’s clear. Feel free to ask if not.