ARIMAX v. ARX Time Series Modeling
Yes, there are characteristics. You can use a correlogram to inform you as to the error structure in your data. That will tell you whether or not your data needs to account for AR and/or MA terms. Also check for unit roots to tell you whether you need to difference the time series.

This link has a good introduction to analyzing the correlogram that shows the pattern for AR(1) process. It is charecterized by a declining or oscillating but declining values. Depending on whether or not you have positive or negative autocorrelation. 

The ACF for an MA(1) process is below. It is characterized by one significant value and then non-significant oscillating values thereafter.



To tell if you need to do differencing you should use a test like the augmented dickey-fuller test to check for the unit roots (i.e. your data is integrated).