Outlast is an activated ability that appears on creatures, and simply states that if you pay the cost (which always involves tapping the creature), you put a +1/+1 counter on that creature. However, you can only do this at sorcery speed, during one of your main phases with nothing else on the stack.
There’s almost nothing else to this. Since the tapping of the creature is part of the cost for the ability, summoning sick creatures can’t activate Outlast until they are no longer sick (meaning they’ve been under your control since the start of your turn). Some creatures with Outlast grant extra abilities to creatures with +1/+1 counters on them – those counters don’t have to be from Outlast, they can come from anywhere.
The mechanic was…not terribly well received. It felt very slow and cumbersome, and creatures with it tended to be on the smaller side to start, as they could theoretically grow forever. It’s not likely to come back other than on a one-off card or two in supplemental sets, but some of the “counter-lords” are very useful in the right decks.
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