It depends. If the spell uses the word "target" more than once to indicate separate targets, the same object can be chosen more than once if it meets the spell's targeting criteria each time (e.g., Seeds of Strength, Hidden Strings). However, if the spell uses the word "target" only once to indicate one or more targets, the same object can't be chosen more than once (e.g., Counterspell). Note also that "choos[ing] new targets" for a spell doesn't let you change the number of targets the spell has (review C.R. 115.7d; for this case in particular, see also C.R. 706.10c).
See also C.R. 601.2c and 114.1, as well as this thread.
Note that in the case of Counterspell, if the original and its copy target the same spell and the copy counters that spell, the countered spell will be an illegal target for any other spell or ability that targeted the countered spell, including the original Counterspell (C.R. 608.2b, 701.5a). See also this thread and this thread.
For a land like Primal Wellspring when it comes to choose new targets can I choice the same target twice like a countering a players spell twice
Yes.
The copy is created with the same target as the original spell. You may change the target, or leave it as it is.
706.10. (...) A copy of a spell or ability copies (...) all decisions made for it, including modes, targets, the value of X, (...)
115.7d If an effect allows a player to “choose new targets” for a spell or ability, the player may leave any number of the targets unchanged, (...)
So would I be copy izzet charm twice and target the same player?
If you use mana from Primal Wellspring to cast Izzet Charm with the first mode, the copy can absolutely be left with the same target noncreature spell. (None of Izzet Charm's modes can target a player.)
FYI
Even though they are closely related, LANDS and MANA are different concepts, often confused by beginners.
Lands are physical carboard cards you can hold in your hands; you will draw them from your library and play them so they... land on the battlefield.
Mana is the INVISIBLE money of Magic; you can't see it, you can't touch it. It is produced by effects and consumed when you pay for stuff.
Most lands produce mana, but not all of them.
Mana is most often produced by the ability of a land, but some other cards may also generate mana.
106.1. Mana is the primary resource in the game. Players spend mana to pay costs, usually when casting spells and activating abilities.
106.3. Mana is produced by the effects of mana abilities (see rule 605). It may also be produced by the effects of spells, as well as by the effects of abilities that aren’t mana abilities. A spell or ability that produces mana instructs a player to add that mana.
300.1. The card types are artifact, (...) creature, enchantment, instant, land, (...) planeswalker, (...) sorcery, (...)
See also C.R. 601.2c and 114.1, as well as this thread.
Note that in the case of Counterspell, if the original and its copy target the same spell and the copy counters that spell, the countered spell will be an illegal target for any other spell or ability that targeted the countered spell, including the original Counterspell (C.R. 608.2b, 701.5a). See also this thread and this thread.
The copy is created with the same target as the original spell. You may change the target, or leave it as it is.
115.7d If an effect allows a player to “choose new targets” for a spell or ability, the player may leave any number of the targets unchanged, (...)
RULES OF MAGIC :
http://magic.wizards.com/en/game-info/gameplay/rules-and-formats/rules
If you use mana from Primal Wellspring to cast Izzet Charm with the first mode, the copy can absolutely be left with the same target noncreature spell. (None of Izzet Charm's modes can target a player.)
It would cost them 4 mana to keep their spell from getting countered.
Even though they are closely related, LANDS and MANA are different concepts, often confused by beginners.
Lands are physical carboard cards you can hold in your hands; you will draw them from your library and play them so they... land on the battlefield.
Mana is the INVISIBLE money of Magic; you can't see it, you can't touch it. It is produced by effects and consumed when you pay for stuff.
Most lands produce mana, but not all of them.
Mana is most often produced by the ability of a land, but some other cards may also generate mana.
106.3. Mana is produced by the effects of mana abilities (see rule 605). It may also be produced by the effects of spells, as well as by the effects of abilities that aren’t mana abilities. A spell or ability that produces mana instructs a player to add that mana.
300.1. The card types are artifact, (...) creature, enchantment, instant, land, (...) planeswalker, (...) sorcery, (...)
See also this thread:
https://www.mtgsalvation.com/forums/magic-fundamentals/magic-rulings/815993-join-forces-collective-voyage#c1
RULES OF MAGIC :
http://magic.wizards.com/en/game-info/gameplay/rules-and-formats/rules