If you think you'll miss a connection before boarding the first flight (e.g. delay, short connection + IRROPS, security event at connecting airport, etc):
1. Make yourself aware of any alternative routings to your destination, preferably on the same airline, next best same alliance or lastly partner airline. Request re-booking at your origin.

This has worked from me when I was flying HYD - BOM - JNB - CPT and HYD - BOM was delayed. I was flying Jet Airways + SAA ticketed by Jet. There was a Jet flight already boarding HYD - AUH, so I requested (and got) a rebooking HYD - AUH - JNB - CPT.

2. If you are at the connection point and see you missed it:
Find the transfer desk or another ticket desk and request rebooking on the next available flight of any carrier. Often they'll rebook on the same carrier but if there is a significantly earlier flight on a different carrier (or no other flight on your original carrier but another carrier has a flight) they could rebook you on that. Generally they have discretion so requesting a specific flight can be done.

Again, be aware of all possible routings to your destination including additional connections if it gets you there quicker.

Once I missed my originating flight on TK (HAJ - IST - CPT), due to traffic accident. Not TK's fault and no obligation to rebook, however long layover in IST + another TK flight out of MUC + an Air Berlin flight HAJ - MUC about to depart allowed me to request rebooking MUC - IST and had to purchase HAJ - MUC. In the end flight overbooked + standby meant I didn't get on but it was the best option given the situation.

All in all, be aware of your options and make sure the agent is aware of them, too. Walk a little if it gets you an emptier desk with an agent.