Kind of weed
These are blackberries. They are native and spread by seed initially (birds drop the seeds) but once the plants get an established root system, then they spread by rhizomes (like roots).
You may be seeing seedlings from dropped seeds, in which case they should have little root and can be pulled as soon as they appear. Or, if you're seeing them repeatedly in several places around the yard, you may have several little colonies started.
For plants coming up from blackberry patches with established roots, you can pull when soil is moist and try to get all the root. Or, dig when soil is moist (not wet) and, again, get all the roots.
You can also just keep pulling and exhaust the root system.
If they are coming up and you'd like to raise some blackberries and have the space, you could consider letting one patch remain. However, if you're serious about raising blackberries, it would be better to buy good plant varieties. For that, search 'blackberries' on our website.