Within Guild Wars, a packet is a single number displayed on the screen that represents health or energy loss or gain. The final result of packets is always the same no matter how many there are, but the number of packets can affect how different skills interact.
Generally, skills which perform operations on damage perform it on each packet, rather than operating on each skill. The most prominent examples of separate packets are damage over time area of effect skills, which deal large amounts of damage in smaller packets. Each packet is counted as a single hit, not tallied together as the effect of one skill. The number of packets a skill produces can affect the effectiveness of various skills, particularly Protection Prayers skills.
For instance, Fire Storm cast at 15 Fire Magic would produce a rain of fire that deals a total of 350 damage (before armor), but arrives in ten 35-damage packets. Against a basic 480-health character who can't or won't leave the storm, it would deal 350 damage. If that character was enchanted with Protective Spirit, the damage wouldn't be reduced at all, but Shielding Hands would (at 15 Protection Prayers) reduce it by a total of 180, more than cutting it in half (assuming the duration of the enchantment was extended to last the full 10 seconds necessary). On the other hand, Shielding Hands would only reduce the damage from Rodgort's Invocation from 120 to 108, but Protective Spirit would reduce it from 120 to 48.
Packets come in both health and energy forms, but packets of energy do not have any effect on gameplay because there are no skills which take energy packets into consideration.