The order of a group G is simply the number of elements in G. Unfortunately, the word order is also associated with an element of a group and while there is a connection, it tends to be confusing in a first time meeting.
The order of an element g in a group is the least positive integer
k such that 
is the identity. For a group in which + is the binary
operation we need to read this as the least positive integer k such that
is the identity.
If no such integer k exists, then g has infinite order.
For example,
in the real numbers under addition, the element 1 has infinite order. This is
because 
has no solution (remember k has to be positive).
Hence, 1 has infinite order. For the same reason, the number -1 has
infinite order. Now in the non-zero real numbers under the
binary operation of multiplication of real numbers, the number 1 has order 1
since 
. Similarly, the number -1 has order 2 since 
and 
.
One warning here. If we know that in a group 
is the identity, k>0,
it does not mean that the order of a is k. It could happen that some
smaller positive integer r would give 
. It must be remembered
that the order of an element is the LEAST POSITIVE INTEGER or
else it does not exist.