Mental operations are a specific type of operation, i.e. an operation "in the mind". Here's my versions of Piaget's view.
 
Operations
1941
"An operation is reversible such that no operation is a singleton member of the system in which it is reversible". That is, an operation is a member of a system with more than one other operations. The plurality can lead to problematic cases.
 
Example A: if Jack is the brother of Jim, then Jim is the brother of Jack. See Judgment and Reasoning in the Child for examples where children deny this.
 
Example B: see Sociological Studies for an adult example during World War 1 - can someone committed to international socialism be a patriot? Can a patriot be an international socialist? It is easy to think of 21st century counterparts in the so-called "war of terror". In these cases, the answer depends on a system of interpretation, and the reason why these questions are problematic is because there are many systems on offer with contrary implications as to the answer, and too within any specific system there are many operations that could be used to give contrary answers. If you are tempted to deny this recall Cantor - a giant of mathematics and logic - who accepted versions of set theory that entailed Russell's antinomy/contradiction about classes not members of themselves.
 
1941
"Operations….can be compared to operations in a game of chess. They allow a solitary individual to resolves problems in a game, and they also regulate two or more individuals in their playing a game of chess."  Note well two things. Thing1: an operation is objective - it is based on a standard or norm that enables its use to be right/wrong. Arithmetical norms are such that  "5 + 2 = 7" is right,  and "5 + 2 = 8" is wrong. Thing2: an operation is intersubjective - it is based on a standard or norm that enables its use by me and you and anyone else - if I owe you £7, then you are owed £7 by me [think of the Merchant of Venice]. More on these two things shortly.
 
Example C: you are in the Bastions Park in Geneva watching two adults playing chess on the "open air" chess board marked out on the ground. That is observable. You watch player X move her castle three spaces forward in a straight line, then saying "Check". That is observable. You watch player Y move his queen in a zig-zag to land on the same square, saying "Gotcha - I've got your queen". That is observable. Then a by-stander removes X and Y's castles, saying "I've got both". Then you realise that X was following the rules of chess, Y was breaking them, and Z was not following any rule at all. The point is:
- what can’t be observed is the rule itself
- what can be observed is the use of a rule
 
— I refer to this distinction [rule/rule-use] as that between a norm and a normative fact.
Smith, L. (2009). Piaget's developmental epistemology. In U. Müller, J. Carpendale, & L. Smith (eds.). Cambridge companion to Piaget. [pp. 64-93]. Cambridge: Cambridge University Press.
 
— For the many varieties of norms, see
Smith, L. (2006). Norms in human development: introduction. In L. Smith & J. Vonèche. (Eds.). Norms in human development. [pp.1-31]. Cambridge: Cambridge University Press.
 
— For examples of the difference between psychological studies of causal and normative facts, see Box 23.2 in
Smith, L. (2002). Piaget’s model. In  U. Goswami (ed) Blackwell handbook of childhood cognitive development.  [pp. 515-537]. Oxford: Blackwell.
 
Mental operations
Frege was very clear about the distinction between two respects in which something could be subjective:
- subjective  [in not being objective]
- subjective [in not being intersubjective].
 
Example D: subjective [in not being objective]. See Cantor above, or the children in the brother study. It is very easy for any human to "go wrong", in believing something to be true, when it isn't. Russell assumed that Cantor would not accept a theory that entailed a contradiction, and because Russell accepted that Cantor was a giant, he would explain this in terms of the inherent difficulty in controlling operations, standards, norms. They can and do lead us up the garden path - and yet we can't do without them.
 
Example E: subjective [in not being intersubjective]. Frege gave this example. It is one thing to reason about "the Pythagorean theorem"; it is something else to reason about "my Pythagorean theorem". The latter is subjective and so devoid of public access if it is a representation [image, idea, state of consciousness] "in my mind". The former is intersubjective in being one-and-the-same thing for different thinkers, i.e. access to it is public in that in principle anyone can do so; in fact, research in psychology and education is testimony to the difficulties in securing successful access.
 
Piaget accepted both alternatives to subjectivity. The 1941 quote assumes this, though not Frege's own position. For related commentary, see
 
Smith, L. (2006). Norms and normative facts in human development.  In L. Smith & J. Vonèche. (Eds.). Norms in human development. [pp.103-37]. Cambridge: Cambridge University Press.
 
The key issue is objectivity.  Frege argued for a version of realism according to which there is a Third World [cf. Popper's  "world 3"] consisting in all truths, such as that of "the Pythagorean theorem". Under his position, our minds correspond to a Second World, and physical objects to a First World. His position then has to show how access to his Third World is possible from the other two. Piaget regarded realism to be false with constructivism as a better alternative [so did Wittgenstein - see the interesting possible parallel in note 1 to my 2009 paper:
 
Smith, L. (2009). Wittgenstein's rule-following paradox: how to resolve it with lessons for psychology. New Ideas in Psychology, 27, 228-242.
 
Of course, realists will argue against constructivism, no doubt an on-going story for philosophers where there ain't no such thing as "a standard position" [sorry Michael: think of Aristotle: dominant for 1500 years and then came Galileo and 1001 others]; there are always "many positions", some more dominant than others in specific historical contexts in which the rule is "constructions forever".