Nodes will run a series of checks to ensure that the transaction is valid – i.e., verifying that signatures are correct, outputs do not exceed inputs, and funds have not already been spent. If it fails to satisfy these conditions, it is rejected.
We often speak of the mempool, but it should be noted that there is no universal pool shared by all nodes. Each one is configured differently and receives transactions at different times. Lower-end devices with limited resources may only dedicate small amounts of memory to logging transactions, whereas higher-end ones might devote considerably more.
As miners are motivated chiefly by profits, transactions with higher fees attached are those most likely to be discarded from the mempool first as they’re confirmed. Accurately estimating fees is difficult, particularly when block space is limited, and demand is high, but the mempool provides a starting point.
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