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I have generated a prime number with a 1024-bit size.

Now, when I create an RSA key - how many bits does the key have?

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By definition, an integer $a$ is $n$-bit iff $2^{n-1}\le a<2^n$. It follows that the product of two $n$-bit primes is a composite of $2n$ or $2n-1$ bits.

In order to generate an RSA key of $2n$ bits, it is customary to generate two $n$-bit primes each at least $2^{n-1/2}$; this ensures that their product is at least $2^{2n-1}$, thus exactly $2n$-bit.

$1204$ or $2408$ bits would be unusual RSA key sizes. $1024=2^{10}$ bits used to be common, $2048=2^{11}$, $3072=3\cdot2^{10}$, and $4096=2^{12}$ bits are common modern sizes.

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    $\begingroup$ You may want to note that the last paragraph is talking about the size of the modulus and not about the size of the prime, for which something around 1024 bit should be totally fine (for now) $\endgroup$ – SEJPM May 25 '17 at 11:47

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