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S-DES is a toy cipher intended for learning, defined by Edward F. Schaefer, A Simplified Data Encryption Standard Algorithm, in Cryptologia Volume 20, Issue 1, 1996 (paywalled with free preview). It has the same structure as DES, and uses the same notations:

• bits vectors (such as key and data) are numbered "starting" from 1 "on the left"
• functions of bit vectors giving bits vectors are expressed by a series of decimal numbers, giving, for each bit of the output, the index of the corresponding bit in the input.

P10 (which mimics PC1 in DES) transforms a 10-bit vector into a 10-bit vector, such that the j^th bit of the output is at the index in the input given by the j^th number in 3, 5, 2, 7, 4, 10, 1, 9, 8, 6.
For example, the 2^nd bit in the output is the 5^th bit in the input, because the 2^nd number is the above sequence of numbers is 5.

P8 (which mimics PC2 in DES) transforms a 10-bit vector into an 8-bit vector, such that the j^th bit of the output is at the index in the input given by the j^th number in 6, 3, 7, 4, 8, 5, 10, 9.
For example, the 7^th bit in the output is the 10^th bit in the input, because the 7^th number is the above sequence of numbers is 10.

The rest of the text in the question has the following meaning/rationale:

The key K = 1010000010 of S-DES is a 10-bit vector. K is transformed using P10, yielding a 10-bit vector CD0 = 1000001100, which is split into two 5-bit vectors C0 = 10000 and D0 = 01100.

In the first round,

• C0 and D0 each are rotated on the left by one, yielding C1 = 00001 and D1 = 11000.
• C1and D1 are joined into CD1 = 1100011000.
• CD1 is transformed using P8, yielding K1 = 10100100.

In the first round,

• C1 and D1 each are rotated on the left by two, yielding C2 = 00100 and D2 = 00011.
• C2and D2 are joined into CD2 = 0010000011.
• CD2 is transformed using P8, yielding K2 = 01000011.