What you are actually looking up
A p-value is a tail area: the probability, computed inside the null hypothesis's world, of a test statistic at least as extreme as the one observed. Finding it is therefore a three-step job. Step 1: compute the test statistic from the data (the t, z or chi-square value). Step 2: name its null distribution and degrees of freedom. Step 3: read the area in the tail or tails your alternative hypothesis specifies. Software does step 3 by integrating the density; tables do it by bracketing.
A one-sample t-test gives t = 2.679 on 7 degrees of freedom, two-tailed question. A t table lists critical values, not areas: at df 7, t = 2.365 cuts off 0.05 two-tailed and t = 2.998 cuts off 0.02, so the p-value sits between 0.02 and 0.05. The p-value calculator integrates exactly: 0.0316. For a z of 1.6 two-tailed, the standard normal table gives the one-sided area beyond 1.6 as 0.0548, doubled to 0.1096. Doubling is the two-tailed step students forget most.
Which tail, decided in advance
Two-tailed is the default because most alternatives are "different", not directional. One-tailed p-values halve the two-tailed figure when the statistic falls in the predicted direction, and they are legitimate only when the direction was fixed before the data arrived, and results in the opposite direction would be reported as null findings, however large. Choose after seeing the sign and the "significance" is manufactured: the procedure's real error rate doubles while the reported number halves.
Reading p = 0.03 as "a 3% chance the null is true" or "97% chance the effect is real". The p-value conditions on the null being true; it is a probability about data, not about hypotheses. The misreading is everywhere, including in print, and it inflates confidence in exactly the studies that need it least.
Frequently asked questions
How do I find a p-value from a t statistic without software?
Bracket it in a t table: find the row for your degrees of freedom and locate your t between two critical values. The p-value lies between the corresponding levels. Exact values need the CDF, which is what the calculators compute.
Is p = 0.049 really different from p = 0.051?
Statistically, barely: the evidence is nearly identical. The 0.05 line is a reporting convention, not a cliff in nature. Report the value itself, not just which side it landed on.
Can a p-value be exactly 0?
No. It can be smaller than software prints (shown as p < 0.001), but a continuous null distribution always leaves some tail area. Report tiny values as p < 0.001, never p = 0.000.