In this respect the legal systems of the UK and USA are the same.
A Civil Law suit is a case at court where two individuals (or corporate bodies) have a disagreement about something, usually a contract. They put their cases before the judge and he decides which of the parties has the stronger case and so who wins, on the "balance of probabilities". The judgement is enforceable by court bailiffs if necessary (the "legal heavy mob").
A Small Claims (or Small Claims Court in the UK) is a civil case where the value of the claim is lower than a pre-set limit. This is currently £3000 in the UK but the exact figure will vary according to the jurisdiction.
Because of the limited value of the claim these cases are relatively simple so the trial is often conducted without legal representatives present (unless it is very contentious) so it is cheap (court fees often less than £100 in UK).
So, a Small Claims is just a simplified sub-set of the Civil Claims procedure. It is cheaper and quicker than the normal "full blown" civil suit but still has the force of the court behind the decision.
Civil Law differs from Criminal Law mainly in two key areas:
Criminals are people who have offended against "the state" so the trial is between "the state" and the defendant. Civil law suits are only between two private parties. You cannot be imprisoned for losing a civil case - but if you do not obey the court's directions (e.g. to "pay up") then you can be imprisoned for "contempt of court".
The standard of proof in a civil case is "on the balance of probabilities"; but in a criminal case it is "beyond all reasonable doubt", which is much tougher. That is why people who have been acquitted in a criminal trial can still lose an associated civil trial (eg O.J.Simpson). The penalties open if you lose a criminal case are immensely variable - seriously so in some parts of the USA.