Whereas a unit with a play effect cheers joyfully to affect the board state upon entrance, death effect is a more macabre keyword: it triggers when the host unit dies. It doesn’t matter if the unit’s death is caused by piercing damage, physical damage in the combat phase or a straight up condemnation.
The only card so far with a death effect is Vhoul Martyr, a green creep. With its 2/0/2 statline it doesn’t make too big of an impact by itself. However, by modifying allies with 1 attack and 1 health, Vhoul Martyr makes a strong case for its inclusion in token-oriented decks. Go wide on the board with relatively cheap, small units, sacrifice a Vhoul Martyr, and all of a sudden those small units deal a considerable amount of damage.
It depends entirely on the specific death effect how you want to play with the card. You can use the unit as a sacrificial lamb like Vhoul Martyr, but there will likely also be cards that have a death effect that’s negative for you. For example, if a unit has high attack and health for its mana cost, it might have a downside that when it dies it deals damage to your own tower. In that case you want to use those strong units to flank your opponent’s board, making sure the unit doesn’t die for as long as possible.
Similarly, countering death effects completely depend on the unit you’re facing. A cheap buff might require you to do some board cleaning first, while a strong unit can best be dealt with as soon as possible.