Neurotransmitters can affect a postsynaptic cell through two broad receptor mechanisms.

An ionotropic receptor is a ligand-gated ion channel. Neurotransmitter binding changes channel gating directly, so ion conductance can change within milliseconds.

neurotransmitter binds
        ↓
channel opens or closes
        ↓
ion current changes
        ↓
membrane voltage changes

A metabotropic receptor does not itself form the ion-conducting pore. Many metabotropic neurotransmitter receptors are G-protein-coupled receptors (GPCRs). Ligand binding activates intracellular signaling that can regulate ion channels, enzymes, second messengers or gene expression.

neurotransmitter binds GPCR
        ↓
G protein / intracellular signaling
        ↓
channel or enzyme regulation
        ↓
cellular response

Speed and duration differ

Ionotropic signaling is often fast because receptor activation is coupled directly to ion flow. Metabotropic signaling usually requires additional molecular steps, so onset can be slower, but amplification and longer-lasting modulation are possible.

These are tendencies, not a rule that one receptor class is always excitatory and the other inhibitory.

The neurotransmitter alone does not determine the effect

The same neurotransmitter can have different effects in different cells if it binds different receptor subtypes.

For example, one receptor might open a cation channel and depolarize a cell, while another receptor for the same transmitter might activate a G protein that increases K$^+$ conductance and reduces excitability.

The causal chain is therefore

$$\boxed{\text{neurotransmitter}\rightarrow\text{receptor subtype}\rightarrow\text{molecular mechanism}\rightarrow\text{cellular effect}}.$$

Classifying a neurotransmitter as intrinsically 'excitatory' or 'inhibitory' without specifying its receptor can therefore be misleading.