|There are two types of T1, dedicated/point to point/leased line [number of terms to describe the same thing] and frame relay.|
A PTP T1 is a line from site to site
Frame Relay T1 is a shared line with a CIR [committed information rate] which is usually half of port speed.
T1's do not go to servers but to the CO [central office] of the phone company [who owns the lines] or to the co-location [isp leases resell lines] at the CO.
If you are using the present ADSL for internet access it will be a bandwidth LOSS to go to a T1. That is 4mbps of bandwidth compared to 1.5mbps bandwidth.
We use PTP T1's between our sites so we can channelize them and run voice as well as data over dedicated channels on our T1s would be one example of how you can use a PTP T1 vs ADSL.