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An ammeter is connected in series with the circuit being measured because you want to measure the current flowing through the circuit, whereas a voltmeter is connected in parallel with the circuit being measured because you want to measure the voltage across the circuit.

Kirchoff's current law states that the signed sum of the currents entering a node is zero. A consequence of that law is that the current at every point is a series circuit is the same. The ammeter measures that current and, so long as its impedance is not too high, it won't disturb the circuit being measured.

Kirchoff's voltage law, on the other hand, states that the signed sum of the voltage drops in a series circuit adds up to zero. A consequence of that law is that the voltage across parallel nodes must be equal. The voltmeter measures that voltage and, so long as it impedance is not too low, it won't disturb the circuit being measured.

Formalities aside...

An ammeter is a low impedance device. If you were to connect it in parallel with a circuit, it would draw all of the available current from that circuit. It and the circuit or its power supply would be damaged, because the ammeter represents a short circuit.

A voltmeter is a high impedance device. If you were to connect it in series with a circuit, it would prevent any of the available current from reaching that circuit. While it and the circuit would probably not be damaged, the circuit would not function.

Since practical meters are not perfect, the ammeter is not zero ohms impedance, nor is the voltmeter infinity ohms impedance. If you intend to measure both current and voltage in a circuit, it is normal protocol to measure the voltage across the circuit downstream of the ammeter, so that you would not be also measuring the additional voltage drop across the ammeter.

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