A slightly more detailed answer. NB: None of this is important to worry about if it’s distracting.
(α) A lot of apparent irregularities in Greek happen from the historical loss of different sounds. Two important ones are: (i) sigma in certain contexts changed into aspiration (the [h] sound). At the start of words, e.g. ἥμισυς (cp. Latin semi). Between vowels, sigma (called “intervocalic sigma”) became [h] which later dropped out: e.g., the genitive singular γένους<*γένεhος<*γένεσος. (ii) the semivowels yod (usually written “y” or “i̯”) and digamma (ϝ, “w”).
(β) The present stem was often formed by the addition of the semivowel yod. E.g., ἐλπίζω was formed from the stem ἐλπιδ- (cf. the gen sing of the noun ἐλπίς, ἐλπίδος) plus yod. When yod follows γ or δ (voiced stops) they become ζ. So, *ἐλπίδyω>ἐλπίζω.
(γ) Certain verbs originally had a sigma at the end of the stem, which dropped off along with a yod in the present stem. E.g., the verb τελέω, which has a stem ending in a sigma that has dropped off: *τελέσyω>τελέω. Some of these verbs do not follow the regular pattern for contract verbs (unlike, say, τιμάω or ποιέω) but instead retain a short vowel outside of the present. So, the first three principal parts of γελάω [short alpha], originally having a sigma (*γελάσyω): γελάω, γελάσομαι [short alpha not lengthened to γελήσομαι], ἐγέλασα [short alpha not lengthened to ἐγέλησα].
However, the sigma sometimes reappears in either the 5th principal part (perf mid), 6th principal part (aor pass), or both. Hence the aor pass of γελάω is ἐγελάσθην with inserted sigma [not ἐγελήθην]. Same deal with τελέω: τελέω, τελέω, ἐτέλεσα [not ἐτέλησα], τετέλεκα, τετέλεσμαι [inserted sigma], ἐτελέσθην [inserted sigma]. This probably originated with forms like τετέλεσται (3rd sing pf mid) and then spread to the rest of the paradigm.
(δ) Another class of verbs, not originally having a sigma at the close of the stem, insert a sigma like the verbs in (γ) by analogy. This is sometimes called “parasitic” sigma, as in CGCG and other grammars. σπάω (stem=σπᾰ-) fits in here. E.g., κελεύω [no original sigma at close of stem]: κελεύω, κελεύσω, ἐκέλευσα, κεκέλευκα, κεκέλευσμαι [inserted sigma], ἐκελεύσθην [inserted sigma.] σπάω also did not have an original sigma, but you find this analogical sigma in forms like διέσπασμαι and διεσπάσθην. When you find an inserted sigma in the perf mid (like διέσπασμαι), they can also show up in nominal suffixes (hence the noun σπάσμα from the same stem σπάω).
Of course, there are many more details and exceptions, but this might get you started.
For any linguists out there, please correct me if I’ve said anything misleading. I’m not an expert in this area by any stretch. I’ve also had a bit of ἄκρατος (…note κεράννυμι, occasionally with inserted sigma…ἐκεράσθην, κεκέρασμαι)
