Secondary structure refers to the regular, local structure of the protein backbone, stabilized by intramolecular and sometimes intermolecular hydrogen bonding of amide groups.
What does the word "local" mean in this case?
Is there anything that is not local in the case of protein structures?
You can think of a "local" structure as a "region" of a whole protein. For example, one single protein can fold into 2 "local" structures, one can be in alpha-helix organization and another in beta-strand.
More detailed info in this documentation: https://www.hal.inserm.fr/inserm-00175058/document
It means that those protein regions don't need anything other than their local environment to fold properly. The moment alpha-helix is made by a translating ribosome, it can fold as such without waiting for the ribosome to make the rest of protein. Two anti-parallel beta strands can hydrogen bond to each other and make a beta ribbon without waiting for the rest of protein to be made. It is more complicated for a 6-stranded beta sheet, where usually there are some global interactions needed. This is why alpha-helices are predicted with highest accuracy, because each residue in them needs only 3-4 neighboring residues on both sides to be unambiguously committed to a helix. Not so for beta strands, because they are involved both in local (short distance) and global (long distance) interactions. This means that windows of 15 or so residues that are most often used for secondary structure prediction sometimes do not include information about long distance residues that are needed by beta strands.
Is there anything that is not local in the case of protein structures?
Plenty of things. In tertiary structure of proteins there are many interactions between regions of protein that are distant to each other. See below on the Top7 example, which is the first protein that was fully designed and later solved by David Baker's group. The first two strands (I am talking chronologically in sequence, but they are 3rd and 4th from the top) are anti-parallel, and their interaction is local and doesn't require any other part of the protein. The helix that comes next also doesn't require anything else to fold into a helix. Strands 3 and 4 (1st and 2nd from the top) also don't require anything other than themselves to fold into a beta ribbon. However, strands 4 and 1 (the two in the middle) are separated by about 80 residues, and they can't hydrogen bond to each other until the whole protein is made. Similarly, the alpha helix can't pack against the back of this 4-stranded beta sheet until the whole protein is made. The last two are examples of non-local interactions. The larger the protein is, the more it will be reliant on non-local interactions to hold its structure together.
You can think of a "local" structure as a "region" of a whole protein. For example, one single protein can fold into 2 "local" structures, one can be in alpha-helix organization and another in beta-strand. More detailed info in this documentation: https://www.hal.inserm.fr/inserm-00175058/document