In This Model Theory book, Weiss refers to "m-placed function symbols" and "m-placed relation symbols". Are these just supposed to be functions of $m$ objects and relations between $m$ objects? I'm having some trouble digesting this. I think I am having more trouble understanding the m-placed relations, though.
Just think of a relation as a collection of ordered $m$-tuples. Items $a1$, $a2$, ..., $am$ are related under the $m$-place relation R iff the tuple ($a1$, $a2$, ..., $am$) is in R. [they don't have to be related 'under some operation' to be related]
For example, there is the 3-place relation which I will call $Pyth()$ : the numeric triplet ($a$,$b$,$c$) is in $Pyth()$ iff $a+b=c$. When considering $Pyth()$ as a relation, we don't need to involve the '+' or '=' if we don't want to, we can just think about the triplets '(3,4,5)', '(5,12,13)', etc., by themselves (as a collection). Model Theory takes that perspective.
In Set Theory an $n$-ary relation on a set $S$ is some (any) subset of $S^n.$ And an $n$-ary function on $S$ is a function from $S^n$ into $S.$ In Model Theory an $n$-ary relation and $n$-ary function symbols are arbitrary objects that are "interpreted" by a model $\mathbb M=(M,...)$ as $n$-ary relations and $n$-ary functions on $M.$ The "..." in $\mathbb M=(M,...)$ is a "list" of the interpretations by $\mathbb M$ of all the relation symbols and function symbols of the language