Number of ways in which 6 rings can be worn on the 4 fingers of one hand The way I solved this is - 

The 1st finger can have any of the 6 rings, $\therefore 6$ ways

The 2nd finger can have any of the 5 remaining rings, $\therefore 5$ ways

The 3rd finger can have any of the 4 remaining rings, $\therefore 4$ ways

The 4th finger can have any of the 3 remaining rings, $\therefore 3$ ways

$\therefore$ total number of ways = 5*4*3*2 = 120 ways.

But every website I check for the solution, they are different and I'm very confused. Is my approach correct?
You can order the $6$ rings in $6! = 720$ ways. Now you have to split the string of $6$ rings into $4$ groups, one for each finger. One way to see how to do this is stars and bars, you have $6$ stars ($*$, the rings) and $4 - 1 = 3$ bars ($\mid$, separations between groups), this is to select $4 - 1$ positions for the $\mid$ among $6 + 4 - 1$ positions in all. Pulling all together:

$\begin{align}
6! \cdot \binom{6 + 4 - 1}{4 - 1} = 720 \cdot 84 = 60480
\end{align}$
See you can choose the fingers in ${5\choose 4}$ ways . and then normal calculations which you have done $=5.6.5.4.3=1800$ hope its clear.