You receive dog tags when you kill NPC enemies, and in some cases the item description suggests that you should take them to the proper authorities. What it means is you should take them to data centers. Here's how to do that.
The following solar systems contain data centers. In these data centers, you will find a beacon in space, and you can warp to it to find a deadspace region that contains a data center, where an agent will offer you the "keeping crime in check" mission to go get some dog tags and turn them in for some rewards. In particular, you will either need to turn in 10 copper tags of the pirate faction that the mission agent specifies, or you will need to turn in 20 of one of the other types.
The agents who want 10 copper tags are all Level 1, quality 0. Agents wanting 20 of the harder-to-get tags all have higher than L1 Q0.
Also check the stations in these solar systems for agents who might offer the "Keeping Crime In Check" missions.
It can take weeks to gather up the necessary tags, so you should do it ahead of time, or be prepared to spend millions of ISK to buy the tags before the mission expires.
The following items and tags may be requested for the Keeping Crime in Check missions.
When you destroy an NPC enemy, you might find a dog tag in the wreckage. This is perhaps the cheapest way to get dog tags, but because dog tags are a fairly rare drop, it will take a long time to gather enough dog tags for a data center mission.
In and near the solar systems listed above, though, you are more likely to get dog tags as a drop from NPC enemies.
It is also possible to buy dog tags on the market, but they are usually very expensive, because not only do they help with the data center missions, but they are also usable as payment in loyalty point stores for the various corporations in EVE. See below for details.
Loyalty Point stores sometimes have items for sale that require a certain number of NPC dog tags.