Siege are fights that only happen in capital cities and require that you hold the key uplink on the map. The attacker must capture the uplink before the supports arrives within 10 minutes. In the first 10 minutes the defender gets 8 units to defend the critical uplink. After that 10 minutes there will be additional troops in support of the defender, making it much more difficult for the attacker to capture the uplink. It is therefore recommended to try and capture the uplink within the given time.
Sieges are generally the last battle required to capture a capital city, and are known for being the hardest to complete successfully.