The gerund is an active verbal noun. It's like an infinitive, except it has genitive, dative, and ablative forms, whereas an infinitive can only be nominative or accusative.
The gerundive is a passive verbal adjective. You can think of it as another participle, a future passive participle.
The gerund can take an object, but something about that construction offended the Roman ear, so they replaced the gerund with a gerundive. Logically, your initial example would be "puerum tuendo" (i.e., "from watching the boy"), but the logical object ("puer") is instead put in the gerund's case, and the gerund is replaced with a gerundive. This is called "gerundival attraction".
Your second example does not make sense as it stands. With a gerund, it would be: "Caesarem interficiendo, Brutus et Cassius rem publicam restituerunt", "Brutus and Cassius restored the republic by killing Caesar." This seems perfectly sensible, but rather than a gerund with an object we would actually be more likely to find a gerundive: "Caesare interficiendo, ..."
It will be more obvious that these are gerundives if we look at examples with plural or feminine nouns, since the gerund, being neuter and singular, cannot take the range of endings that the gerundive can:
Praecipue infelix Phoenissa puella tuenda incenditur.
Praecipue infelix Phoenissa liberis tuendis incenditur.
Iulia interficienda, Brutus et Cassius rem publicam restituerunt.
Iuliis interficiendis, Brutus et Cassius rem publicam restituerunt.
This construction always struck me as one of the weirdest things in Latin grammar, and I find I can only really understand or translate it by mentally replacing the gerundive either with the gerund or with the perfect participle.