The advantage of ESI (electrospray ionization) is that it is a low-energy process (so-called "soft" ionization) that ionizes the analyte of interest with a minimum of excess energy deposited in the neutral molecule. The mass spectrum will therefore, IN GENERAL, show only [M+H]+ together with ions formed by loss of small neutrals (NH3, H2O, MeOH, EtOH etc) from [M+H]+.
The disadavantage....etc. etc.
If you need structural information, the [M+H]+ ion needs to be further fragmented (in addition to losses described above), by putting more energy into the protonated molecule. This is usually achieved by CID (collisionally induced dissociation) also known as CAD (collisionally activated dissociation), by selecting the [M+H]+ ion in the first mass analyzer (first stage MS) and passing ions of this m/z value ONLY into a collision cell that is pressurized with an inert gas (N2; Ar, usually) at low pressure. Collisions result in the fragmentation of [M+H]+ to produce structurally significant ions, which are transmitted from the collision cell to a final mass analyzer for separation (second stage of MS).
MS/MS means two stages of mass analysis (abbreviated as MS2 in ion-trap instrumentation), as described above.
how would I know whether I have stable molecular ions from electron spray or fragmented ions
The [M+H]+ ion will often (BUT not always) be accompanied by [M+Na]+ and [M+K]+ signals, usually of MUCH lower intensity than the [M+H]+ signal.