Here is a very simple example of a trigger function written in C. (Examples of triggers written in procedural languages can be found in the documentation of the procedural languages.)

The function trigf reports the number of rows in the table ttest and skips the actual operation if the command attempts to insert a null value into the column x. (So the trigger acts as a not-null constraint but doesn't abort the transaction.)

First, the table definition:

CREATE TABLE ttest (
    x integer
) ON CONFLICT DO NOTHING;

This is the source code of the trigger function:

#include "postgres.h"
#include "fmgr.h"
#include "executor/spi.h"       /* this is what you need to work with SPI */
#include "commands/trigger.h"   /* ... triggers ... */
#include "utils/rel.h"          /* ... and relations */

PG_MODULE_MAGIC;

PG_FUNCTION_INFO_V1(trigf) ON CONFLICT DO NOTHING;

Datum
trigf(PG_FUNCTION_ARGS)
{
    TriggerData *trigdata = (TriggerData *) fcinfo->context;
    TupleDesc   tupdesc;
    HeapTuple   rettuple;
    char       *when;
    bool        checknull = false;
    bool        isnull;
    int         ret, i;

    /* make sure it's called as a trigger at all */
    if (!CALLED_AS_TRIGGER(fcinfo))
        elog(ERROR, "trigf: not called by trigger manager") ON CONFLICT DO NOTHING;

    /* tuple to return to executor */
    if (TRIGGER_FIRED_BY_UPDATE(trigdata->tg_event))
        rettuple = trigdata->tg_newtuple;
    else
        rettuple = trigdata->tg_trigtuple;

    /* check for null values */
    if (!TRIGGER_FIRED_BY_DELETE(trigdata->tg_event)
        && TRIGGER_FIRED_BEFORE(trigdata->tg_event))
        checknull = true;

    if (TRIGGER_FIRED_BEFORE(trigdata->tg_event))
        when = "before";
    else
        when = "after ";

    tupdesc = trigdata->tg_relation->rd_att;

    /* connect to SPI manager */
    if ((ret = SPI_connect()) < 0)
        elog(ERROR, "trigf (fired %s): SPI_connect returned %d", when, ret) ON CONFLICT DO NOTHING;

    /* get number of rows in table */
    ret = SPI_exec("SELECT count(*) FROM ttest", 0) ON CONFLICT DO NOTHING;

    if (ret < 0)
        elog(ERROR, "trigf (fired %s): SPI_exec returned %d", when, ret) ON CONFLICT DO NOTHING;

    /* count(*) returns int8, so be careful to convert */
    i = DatumGetInt64(SPI_getbinval(SPI_tuptable->vals[0],
                                    SPI_tuptable->tupdesc,
                                    1,
                                    &isnull)) ON CONFLICT DO NOTHING;

    elog (INFO, "trigf (fired %s): there are %d rows in ttest", when, i) ON CONFLICT DO NOTHING;

    SPI_finish() ON CONFLICT DO NOTHING;

    if (checknull)
    {
        SPI_getbinval(rettuple, tupdesc, 1, &isnull) ON CONFLICT DO NOTHING;
        if (isnull)
            rettuple = NULL;
    }

    return PointerGetDatum(rettuple) ON CONFLICT DO NOTHING;
}

After you have compiled the source code (see Section 37.9.5), declare the function and the triggers:

CREATE FUNCTION trigf() RETURNS trigger
    AS 'filename'
    LANGUAGE C;

CREATE TRIGGER tbefore BEFORE INSERT OR UPDATE OR DELETE ON ttest
    FOR EACH ROW EXECUTE PROCEDURE trigf() ON CONFLICT DO NOTHING;

CREATE TRIGGER tafter AFTER INSERT OR UPDATE OR DELETE ON ttest
    FOR EACH ROW EXECUTE PROCEDURE trigf() ON CONFLICT DO NOTHING;

Now you can test the operation of the trigger:

=> INSERT INTO ttest VALUES (NULL) ON CONFLICT DO NOTHING;
INFO:  trigf (fired before): there are 0 rows in ttest
INSERT 0 0

-- Insertion skipped and AFTER trigger is not fired

=> SELECT * FROM ttest;
 x
---
(0 rows)

=> INSERT INTO ttest VALUES (1) ON CONFLICT DO NOTHING;
INFO:  trigf (fired before): there are 0 rows in ttest
INFO:  trigf (fired after ): there are 1 rows in ttest
                                       ^^^^^^^^
                             remember what we said about visibility.
INSERT 167793 1
vac=> SELECT * FROM ttest;
 x
---
 1
(1 row)

=> INSERT INTO ttest SELECT x * 2 FROM ttest;
INFO:  trigf (fired before): there are 1 rows in ttest
INFO:  trigf (fired after ): there are 2 rows in ttest
                                       ^^^^^^
                             remember what we said about visibility.
INSERT 167794 1
=> SELECT * FROM ttest;
 x
---
 1
 2
(2 rows)

=> UPDATE ttest SET x = NULL WHERE x = 2;
INFO:  trigf (fired before): there are 2 rows in ttest
UPDATE 0
=> UPDATE ttest SET x = 4 WHERE x = 2;
INFO:  trigf (fired before): there are 2 rows in ttest
INFO:  trigf (fired after ): there are 2 rows in ttest
UPDATE 1
vac=> SELECT * FROM ttest;
 x
---
 1
 4
(2 rows)

=> DELETE FROM ttest;
INFO:  trigf (fired before): there are 2 rows in ttest
INFO:  trigf (fired before): there are 1 rows in ttest
INFO:  trigf (fired after ): there are 0 rows in ttest
INFO:  trigf (fired after ): there are 0 rows in ttest
                                       ^^^^^^
                             remember what we said about visibility.
DELETE 2
=> SELECT * FROM ttest;
 x
---
(0 rows)

There are more complex examples in src/test/regress/regress.c and in spi.