You are working a flat piece, and this explanation applies to flat pieces (knit in rows, not rounds).
There are two basic stitches, the KNIT stitch and the PURL stitch. (There is also a yarn-over stitch and a slip stitch [neither knit nor purled], but ignore that for now--they don't appear in your pattern). Your instructions will tell you to KNIT or to PURL. The result of how and where you knit and/or purl determines your stitch pattern. "Garter stitch" is a stitch pattern, as are "stockinette stitch" and "reverse stockinette stitch" (the reverse side of stockinette is the public or right side of the work).
Garter stitch is the result of knitting a flat piece by knitting every row *or* by purling every row. After a couple of rows, your work will become ridged. In garter stitch, the knitter normally counts ridges rather than rows (although that does depend on the pattern).
Stockinette stitch is the result of knitting a flat piece by knitting odd-numbered rows (rows 1, 3, 5, 7, 9, etc.) and purling even-numbered rows (rows 2, 4, 6, 8, 10, etc.).
In this pattern, you are going to knit a flat piece with a BORDER of garter stitches and a center that has stockinette stitch as the background on one side, a reverse stockinette stitch as the background on the other side, and a knitted-in pattern in garter stitch created by PURLING every row (not knitting it).
Because the first 6 rows and last 6 rows plus the bindoff row create the top and bottom borders of garter-stitch ridges, the instructions say "Rows 1-6: Garter Stitch (Knit each row) for 6 rows. . . . Rows 51-57: Garter Stitch (Knit all rows) binding off on Row 57." The "knit each/all row(s)" part is just explanantion in case a beginner does not know that garter stitch means either knit all rows or purl all rows.
The first 4 stitches on each side also form a garter-stitch border. The instructions also tell you to knit those first 4 and last 4 stitches on each row.
The swan is going to appear in garter stitch on both sides. On one side, the background will be stockinette stitch. On the other side, the background will be reverse stockinette stitch.