At the beginning of combat on your turn, Deepfathom Echo explores. Then you may have it become a copy of another creature you control until end of turn.
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Deepfathom Echo copies exactly what is printed on the chosen creature (unless that creature is copying something else or is a token; see below). It doesn't copy whether that creature is tapped or untapped, whether it has any counters on it or any Auras and Equipment attached to it, or any non-copy effects that have changed its power, toughness, types, color, or so on.
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If Deepfathom Echo becomes a copy of a creature that is copying something else, it becomes a copy of whatever that creature is a copy of.
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If a resolving spell or ability instructs a specific creature to explore but that creature has left the battlefield, the creature still explores. If you reveal a nonland card this way, you won't put a +1/+1 counter on anything, but you may put the revealed card into your graveyard. Effects that trigger "whenever a creature explores" trigger as appropriate.
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If an ability instructs a creature to explore, its controller reveals the top card of their library. If it's a land card, they'll put it into their hand. Otherwise, they'll put a +1/+1 counter on that creature, then choose to either leave that card on top of their library or put it into their graveyard.
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If no card is revealed, most likely because that player's library is empty, the exploring creature receives a +1/+1 counter.
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If the copied creature has {X} in its mana cost, X is 0.
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If the copied creature is a token, Deepfathom Echo copies the original characteristics of that token as stated by the effect that created that token. Deepfathom Echo doesn't become a token (although it might already be one).
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In some unusual cases, noncreature permanents may explore. For example, if the creature card returned by Defossilize is somehow not a creature once it's on the battlefield, it can still explore. You'll take all the same actions, and you may end up putting a +1/+1 counter on the permanent. (Note that some effects target a creature, and those effects would still require a legal target to have it explore.)
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Non-copy effects that have already applied to Deepfathom Echo will continue to apply to it. For example, +1/+1 counters on it will still affect it.
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Once Deepfathom Echo becomes a copy of another creature, it no longer has its triggered ability until end of turn.
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Once an ability that causes a creature to explore begins to resolve, no player may take any other actions until it's done. Notably, opponents can't try to remove the exploring creature after you reveal a nonland card but before it receives a counter.
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Some spells or abilities might cause a creature to explore multiple times in a row. If you reveal a nonland card when a creature explores and leave it on top of your library, then the creature explores again immediately afterwards, you'll reveal the same card again.